Bijective Relative Gabriel Correspondence over Rings with Torsion Theoretic Krull Dimension

Journal of Algebra 243, 644᎐674 Ž2001. doi:10.1006rjabr.2001.8855, available online at http:rrwww.idealibrary.com on

Bijective Relative Gabriel Correspondence over Rings with Torsion Theoretic Krull Dimension Toma Albu1 Bucharest Uni¨ ersity, P.O. Box 2-255, RO-78215 Bucharest 2, Romania E-mail: [email protected]

Gunter Krause 2 ¨ Department of Mathematics, Uni¨ ersity of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada E-mail: [email protected]

and Mark L. Teply Department of Mathematical Sciences, Uni¨ ersity of Wisconsin᎐Milwaukee, EMS Building, P.O. Box 413, Milwaukee, Wisconsin 53201 E-mail: [email protected] Communicated by Kent R. Fuller Received October 10, 2000

A series of results by Asensio and Torrecillas Ž1992, Comm. Algebra 20, 847᎐866., Gordon and Robson Ž1973, ‘‘Krull Dimension,’’ Memoirs of the American Mathematical Society, Vol. 133., Kim and Krause Ž1999, Comm. Algebra 27, 3339᎐3351., 1 This work was started during the stay of the first author at the University of Wisconsin᎐ Milwaukee as a visiting professor in the academic year 1997r1998 and continued while he was a visiting scientist at the University of Manitoba, Winnipeg, Canada, March 1998, and a visiting scientist, with financial support of the Alexander von Humboldt Foundation, at the University of Bielefeld, Germany, August᎐September 1998, and during his stay at The Ohio State University, Columbus, and the University of California, Santa Barbara, as a visiting professor in the academic year 1998r1999. In the final stage of its preparation he was partially supported by Grant 7Dr2000 awarded by the Consiliul Nat¸ional al Cercetarii ˘ ¸Stiint¸ifice din ˆInvat Superior, Romania. He thanks all these institutions for their ˘¸˘amantul ˆ ˆ hospitality and financial support. 2 The second author gratefully acknowledges support from Grant OGP0007261 awarded by the Natural Sciences and Engineering Research Council of Canada.

644 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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Ž1981, Comm. Algebra 9, 1395᎐1426. give information about rings and Nastasescu ˘ ˘ that have Krull dimension or are noetherian relative to a torsion theory. The aim of this paper is to extend these results to rings R having relative Krull dimension with respect to a hereditary torsion theory ␶ on Mod-R such that any ␶-torsion-free right R-module M has nonempty assassinator. Since any ideal invariant hereditary torsion theory has this property in view of a recent result by the authors Ž2000, J. Algebra 229, 498᎐513., these results apply in particular to the commutative case. 䊚 2001 Academic Press

Key Words: hereditary torsion theory; ideal invariant; ␶-Krull dimension; classical ␶-Krull dimension; tame; ⌬-module; ␶-artinian; ␶-noetherian; bijective ␶-Gabriel correspondence; fully ␶-bounded; assassinator; ␶-restricted strong second layer condition.

INTRODUCTION In this paper we investigate various questions concerning rings with ␶-Krull dimension, such as the structure of ⌬-modules and tame modules, the bijectivity of the ␶-Gabriel correspondence, fully ␶-boundedness, evaluation of classical ␶-Krull dimension, etc., where ␶ is a hereditary torsion theory on Mod-R with the following property:

Ž †.

Ass Ž M . / ⭋ for every ␶-torsion-free right R-module M / 0.

We adopt this general setting of a ring with ␶-Krull dimension satisfying property Ž†. as a means of unifying the methods used by Asensio and Torrecillas w5x, Gordon and Robson w8x, Kim and Krause w11x, and w15x to investigate the structure of ⌬-modules and tame modNastasescu ˘ ˘ ules, the bijectivity of the correspondence w E x ¬ AssŽ E ., where w E x is the isomorphism class of an indecomposable injective module E, and the structure of rings whose torsion-free Žinjective. modules satisfy various restrictions that arise naturally from these objects. Numerous results from the aforementioned sources are shown to hold true for the larger class of rings with ␶-Krull dimension that satisfies Ž†.. Our approach is significant since this class of rings includes rings with Krull dimension, ␶-noetherian rings Žhence, ␶-artinian rings and, of course, noetherian rings., and rings with ␶-Krull dimension, where one of the following is true: ␶ is ideal invariant Žhence, commutative rings with ␶-Krull dimension. or w E x ¬ AssŽ E . is bijective when restricted to the set w E x N E is ␶-torsion-free indecomposable injective4 . After presenting in Section 0 the basic terminology and notation used throughout the paper, we introduce in Section 1 another condition, labelled Ž††., on a hereditary torsion theory ␶ on Mod-R and discuss its relationship with Ž†.. In Section 2, we relativize a series of results on the structure of ⌬-injective modules established in w15x for rings with Krull dimension. In

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particular, we show ŽTheorem 2.9. that for a ring R with ␶-Krull dimension and Ž†., the assignment w E x ¬ AssŽ E . induces a bijection between the set of isomorphism classes of ␶-torsion-free indecomposable ⌬-injective modules E and the set of minimal ␶-closed prime ideals of R. Tame modules play an important role in the theory of localizations of noetherian rings. The concept can be extended to general rings, although two variants emerge whose relationship is discussed in Section 3; in particular, we show that they are equivalent for rings with ␶-Krull dimension. The bijective ␶-Gabriel correspondence is introduced in Section 4. It requires that for a ring R the map w E x ¬ AssŽ E . from the set SpecŽMod-R, ␶ . of isomorphism classes of ␶-torsion-free indecomposable injective modules to the set Spec␶ Ž R . of ␶-closed prime ideals is well-defined and bijective. If this condition is imposed on a ring R with ␶-Krull dimension, then Ž†. is an implicit consequence. The main result ŽTheorem 4.6. of this section provides various characterizations of the bijective ␶-Gabriel correspondence for rings with ␶-Krull dimension; it holds, for example, if and only if all ␶-torsion-free modules are tame. In Section 5, we investigate the relationship between the ␶-Krull dimension and the classical ␶-Krull dimension of a ring R with ␶-Krull dimension and bijective ␶-Gabriel correspondence. It is shown ŽTheorem 5.4. that the two dimensions differ by at most 1 and that they are equal whenever R is either right fully ␶-bounded with Ž†. or right ␶-noetherian. Section 6 deals with ⌬-modules over a ring R with ␶-Krull dimension. We prove ŽTheorem 6.6. that if such a ring satisfies Ž†., then every ␶-torsion-free right R-module with finite assassinator Žresp. with ␶-Krull dimension. is a ⌬-module if and only if Spec␶ Ž R . satisfies the right restricted Žresp. ␶-restricted . strong second layer condition and either R has bijective ␶-Gabriel correspondence or R is right fully ␶-bounded.

0. TERMINOLOGY, NOTATION, AND PRELIMINARIES All rings considered are associative with unit element 1 / 0; modules are unital right modules. By ‘‘ideal’’ we shall mean a two-sided ideal. For standard terminology we refer the reader to w14, 17x. If R is a ring, then Mod-R denotes the category of all right R-modules. We often write MR to emphasize that M is a right R-module; L Ž MR ., or just L Ž M ., stands for the lattice of all submodules of M. The notation N F M Ž N - M . means that N is a Žproper. submodule of M. If N is an essential submodule of M, then we write N Fess M. Whenever we want to indicate that X is merely a Žproper. subset of Y, then we shall use X : Y Ž X ; Y ..

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Let M g Mod-R, let X : M, and let Y : R. Set rR Ž X . [  r g R N Xr s 0 4 s annihilator of X in R , l M Ž Y . [  m g M N mY s 0 4 s annihilator of Y in M. Note that l M Ž Y . F M if and only Y is a left ideal of R. If X s  x 1 , . . . , x n4 and Y s  y 1 , . . . , ym 4 , then we write rR Ž x 1 , . . . , x n . and l M Ž y 1 , . . . , ym . for rR Ž X . and l M Ž Y ., respectively. Subscripts are deleted if there is no danger of ambiguity. Set AR Ž M . [  rR Ž X . N ⭋ / X : M 4 . The module M is a ⌺-module Ž ⌬-module. if the poset Ž AR Ž M ., :. is noetherian Žartinian .; i.e., it satisfies the ACC ŽDCC.. An injective ⌺-module Ž ⌬-module. is called ⌺-injecti¨ e Ž ⌬-injecti¨ e .. A subset X : MR is said to be finitely annihilated if there exist finitely many elements x 1 , . . . , x k g X such that r Ž X . s r Ž x 1 , . . . , x k .. It is easy to see that M is a ⌬-module if and only if each of its nonempty subsets is finitely annihilated. A prime ideal P of R is associated with the right R-module M if there exists a submodule 0 / N F M such that P s r Ž N⬘. for any 0 / N⬘ F N. The set of associated prime ideals of M is called the assassinator of M; it is denoted by Ass R Ž M ., or simply by AssŽ M .. If AssŽ M . s  P 4 , then M is called P-primary and, by abuse of notation, we often write AssŽ M . s P. If AssŽ M . s P s r Ž M ., then M is called a P-prime module. Throughout, ␶ s Ž T , F . is a fixed hereditary torsion theory on Mod-R, and ␶ Ž M . denotes the ␶-torsion submodule of a right R-module M. Note that ␶ Ž R . s ␶ Ž R R . is an ideal of R. The letters ␹ and o stand for the torsion theories with torsion classes T s Mod-R and T s  04 , respectively. We tacitly assume that ␶ / ␹ , since our results are either trivial or vacuously true for ␹ . A submodule N of M is ␶-dense Žin M . if ␶ Ž MrN . s MrN; i.e., if MrN is ␶-torsion. It is ␶-closed if ␶ Ž MrN . s 0; i.e., if MrN is ␶-torsionfree. The ␶-closure of N Žin M . is the submodule N s F C N N F C F M, ␶ Ž MrC . s 04 . It is the smallest ␶-closed submodule of M that contains N and also is the largest submodule of M in which N is ␶-dense. The set  D F R R N D s R4 of ␶-dense right ideals is called the Gabriel filter associated with ␶ . Basic torsion-theoretic results can be found in w3, 7, 16, and 17x. We also use the following notation. E Ž M . s E R Ž M . s injective hull of M g Mod-R

w E x s isomorphism class of the injective right R-module E Spec Ž Mod-R, ␶ . s  w E x N E ␶-torsion-free indecomposable injective 4

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Sat ␶ Ž MR . s lattice of all ␶-closed submodules of MR Sat ␶ Ž R . s Sat ␶ Ž R R . Spec Ž R . s set of all prime ideals of R Spec␶ Ž R . s Spec Ž R . l Sat ␶ Ž R . s set of all ␶-closed prime ideals of R Specg Ž R . s  P g Spec Ž R . N RrP is a right Goldie ring 4 N␶ Ž R . s ␶-closed prime radical of R s FP g Spec ␶ Ž R. P Min ␶ Ž R . s set of all minimal elements of the poset Ž Spec␶ Ž R . , : . Ming Ž R . s set of all minimal elements of the poset Ž Specg Ž R . , : . . It is well-known Žsee e.g., w17, Proposition 4.1, p. 207x. that Sat ␶ Ž M . is an upper continuous modular lattice. Note that any P g Spec␶ Ž R . contains an element in Min ␶ Ž R .. A module MR is said to be ␶-artinian Ž␶-noetherian. if Sat ␶ Ž M . is artinian Žnoetherian .. The ring R is ␶-artinian Ž␶-noetherian. if R R is ␶-artinian Ž␶-noetherian .. The ␶-Krull dimension k␶ Ž M . of a right R-module M is the Krull dimension Žor de¨ iation. K dimŽSat ␶ Ž M .. of the poset Sat ␶ Ž M .. Thus, k␶ Ž M . s y1 if and only if M is ␶-torsion, and k␶ Ž M . s ␣ for an ordinal ␣ G 0 if k␶ Ž M . u ␣ and, given any descending chain M ) C1 ) C2 ) ⭈⭈⭈ ) Ci ) Ciq1 ) ⭈⭈⭈ of Ž␶-closed. submodules, k␶ Ž CirCiq1 . - ␣ for all but finitely many i. Note that if ␶ s o Ži.e., all right R-modules are torsion-free., then k␶ Ž M . s K dimŽ M . [ K dimŽ L Ž M .. is the usual Krull dimension of the module M. If k␶ Ž R . [ k␶ Ž R R . is defined, then R is said to be a ring with ␶-Krull dimension. Note that if R and MR have ␶-Krull dimension, then k␶ Ž M . F k␶ Ž R .. A module M / 0 is called ␶-critical if it is ␶-torsion-free with ␶-Krull dimension and k␶ Ž MrN . - k␶ Ž M . for every 0 / N F M. A ␶-critical module M with k␶ Ž M . s ␣ is called ␣-␶-critical. It is easy to see that a ␶-torsion-free module M with ␶-Krull dimension contains a ␤-␶-critical submodule, where ␤ F k␶ Ž M .. On one occasion, we also use the Gabriel dimension, denoted by G dim, of rings and modules. For basic results concerning this invariant we refer the reader to w9, 16x. If I is an ideal of R, then ␶ I s Ž TI , FI . stands for the torsion theory cogenerated by E R Ž RrI .. In contrast, ␶ I s Ž T I, F I . denotes the direct image of ␶ s Ž T , F . in Mod-Ž RrI ., where T I s  M g Mod-Ž RrI . N MR g T 4 . Whenever there is no danger of confusion, we shall denote the induced

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torsion theory ␶ I by ␶ as well. Note that if M g Mod-Ž RrI ., then Sat ␶ I Ž MR r I . s Sat ␶ Ž MR .; so, in particular, k␶ I Ž MR r I . s k␶ Ž MR ., which will be used frequently, without further mention. The torsion theory ␶ is said to be ideal in¨ ariant if, for every ideal I and every ␶-dense right ideal D of R, the right R-module IrDI is ␶-torsion. Obviously, ␶ has this property whenever R is commutative. Characterizations and more information about these torsion theories can be found in w2, Section 6x. Most of the results presented in this paper hold for ideal invariant torsion theories, due to the following result. THEOREM 0.1 w2, Corollary 6.5x. If R is a ring with ␶-Krull dimension, where ␶ is an ideal in¨ ariant hereditary torsion theory on Mod-R, then AssŽ M . / ⭋ for all 0 / M g F. Remarks 0.2. Ž1. For a commutative ring, Theorem 0.1 gives w4, Lemma 1.4x as a corollary. Also, ␶ s o is trivially ideal invariant; so we recover the well-known result, w8, Theorem 8.3x, that AssŽ M . / ⭋ for a nonzero module M over a ring with Krull dimension. Ž2. The hypothesis ‘‘␶ is ideal in¨ ariant’’ is not necessary to ensure that AssŽ M . / ⭋ for all 0 / M g F ; indeed, the latter holds for any right ␶-noetherian ring. It is still an open problem whether this hypothesis can be removed from Theorem 0.1, although it can be weakened to, for example, the assumption that N␶ Ž RrI . is nilpotent for every ␶-closed ideal I of R Žsee w2, Theorem 6.4 and Proposition 5.2x.. Ž3. The condition ‘‘R has ␶-Krull dimension’’ is also not necessary for the assertion in Theorem 0.1; any commutative ring R which has Gabriel dimension Žbut not Krull dimension. provides an example.

1. THE CONDITIONS Ž†. AND Ž††. The purpose of this section is to discuss the relationship between two conditions on a given hereditary torsion theory ␶ on Mod-R that postulate the existence of certain ␶-closed prime ideals attached to ␶-torsion-free right R-modules. LEMMA 1.1. If I is a ␶-closed semiprime ideal of the ring R with ␶-Krull dimension, then RrI is a right Goldie ring. Proof. See w6, Proposition 2x or w16, Proposition 7.3.9x. LEMMA 1.2. If R is a prime ring with ␶-Krull dimension and R g F , then k␶ Ž RrI . - k␶ Ž R . for any ideal I / 0 of R.

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Proof. By w16, Proposition 7.3.9x we have k␶ Ž R . s sup  k␶ Ž RrE . q 1 N E g Sat ␶ Ž R . and E Fess R R 4 . Nonzero ideals of a prime ring are essential as right ideals; so k␶ Ž RrI . s k␶ Ž RrI . - k␶ Ž R .. PROPOSITION 1.3. rian poset.

If R has ␶-Krull dimension, then Spec␶ Ž R . is a noethe-

Proof. Let P1 , P2 g Spec␶ Ž R . with P1 - P2 . Then P2rP1 is a nonzero ideal of the ␶-torsion-free prime ring RrP1 with ␶-Krull dimension; so k␶ Ž RrP2 . s k␶ ŽŽ RrP1 .rŽ P2rP1 .. - k␶ Ž RrP1 . by Lemma 1.2. Consequently, a strictly ascending chain of ␶-closed prime ideals leads to a strictly decreasing sequence of ordinals. Since the latter must be finite, the assertion follows. Now, consider the following conditions for the torsion theory ␶ on Mod-R.

Ž †.

AssŽ M . / ⭋ for e¨ ery nonzero ␶-torsion-free module MR .

Ž †† .

For e¨ ery ␶-critical module CR , there exists 0 / D F C such that rR Ž D . g Spec Ž R . .

Note that both Ž†. and Ž††. are preserved when passing to a factor ring RrI. At first glance, Ž†. appears to be more restrictive than Ž††.; in fact, this is true, as the following example shows. EXAMPLE 1.4. Let F w X 1 , X 2 , . . . x be the commutative polynomial ring in countably infinitely many indeterminates over a field F. Let I be the ideal generated by  X i2 N i s 1, 2, . . . 4 and set R [ F w X 1 , X 2 , . . . xrI. Let x i be the canonical image of X i in R. Then the ideal P generated by  x i N i s 1, 2, . . . 4 is the unique prime ideal of R. Consider the hereditary torsion theory ␶ s Ž T , F . on Mod-R, where T s  M g Mod-R N M has Gabriel dimension4 . We establish that Ž††. holds vacuously by showing that no ␶-critical modules exist. Suppose that there are nonzero ␶-torsion-free modules with ␶-Krull dimension. Choose ␣ ) y1 to be minimal among the ordinals that arise as their ␶-Krull dimensions, and let C / 0 be ␣-␶-critical. For any 0 / N F C we have that k␶ Ž CrN . - k␶ Ž C . s ␣ ; so, by the minimality of ␣ , k␶ Ž CrN . s k␶ Ž CrN . s y1, which means that every proper homomorphic image of C has Gabriel dimension; hence so does C by w16, Corollary 3.4.5x Žin fact, G dimŽ C . F ␥ q 1, where ␥ s sup G dimŽ CrN . N 0 / N F C 4.. Thus, 0 / C g T l F s 0, which is a contradiction.

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Now, R is a commutative ring with cl. K dimŽ R . s 0. Suppose that R has Gabriel dimension, say ␣ . By w9, Theorem 3.4x, ␣ s 1; so R is semi-artinian and hence contains a nonzero simple submodule S s fR. Since P is the only prime ideal, rR Ž f . s rR Ž S . s P; so fx i s 0 for each i G 0. But then the definition of I would force x i to be a factor of f for every i, which cannot be. Therefore, R does not have Gabriel dimension; hence 0 / Rr␶ Ž R . g F. Now, any prime ideal associated with Rr␶ Ž R . would have to be ␶-closed. But the unique prime ideal P of R is ␶-dense, since RrP ( F has Gabriel dimension. Thus, AssŽ Rr␶ Ž R .. s ⭋; so Ž†. fails for R. However, for rings with ␶-Krull dimension no distinction exists between Ž†. and Ž††.. PROPOSITION 1.5. are equi¨ alent.

If the ring R has ␶-Krull dimension, then Ž†. and Ž††.

Proof. Since Ž†. trivially implies Ž††., we assume Ž††. and show Ž†.. Let 0 / M g F. Since R has ␶-Krull dimension, M has nonzero Že.g., cyclic. submodules with ␶-Krull dimension, and these contain ␶-critical submodules. In the set C [  0 / E F M N E is ␶-critical 4 , choose C such that ␣ [ k␶ Ž RrrR Ž C .. is minimal in the set of ordinals  k␶ Ž RrrR Ž E .. N E g C 4 . By Ž††., there exists 0 / D F C with rR Ž D . s P g SpecŽ R .. Note that RrP g F , since RrP ¨ D D . Clearly, D g C and rR Ž C . F P; so k␶ Ž RrP . s k␶ Ž RrrR Ž C .. s ␣ by the minimality of ␣ . Now, let 0 / D⬘ F D and suppose that P - rR Ž D⬘.; so we would get k␶ Ž Rrr Ž D⬘.. - k␶ Ž RrP . s ␣ by Lemma 1.2. Since D⬘ g C , this contradicts the minimality of ␣ . Therefore, rR Ž D⬘. s P for all 0 / D⬘ F D; hence P s AssŽ D . s AssŽ C . : AssŽ M ., as desired.

2. ⌬-INJECTIVE MODULES The aim of this section is to relativize Žwith respect to a hereditary torsion theory ␶ satisfying Ž†.. some results that have been established in w15x for rings with Krull dimension. DEFINITION 2.1. A right R-module M is k␶-homogeneous if M has ␶-Krull dimension and k␶ Ž N . s k␶ Ž M . for any N F M with N f T. The ring R is k␶-homogeneous if R R is k␶-homogeneous. In w13x, a k␶-homogeneous module is called ␶-smooth. LEMMA 2.2. A ␶-torsion-free prime ring R with ␶-Krull dimension is k␶-homogeneous.

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Proof. Let 0 / A F R R . Standard basic properties of ␶-Krull dimension give k␶ Ž A . F k␶ Ž RA . s k␶

ž Ý xA / s sup  k Ž xA. 4 xgR

xgR

␶

s sup  k␶ Ž Ar Ž r Ž x . l A . . 4 F k␶ Ž A . ; xgR

so k␶ Ž A. s k␶ Ž RA.. Since RA is a nonzero ideal, k␶ Ž RrRA. - k␶ Ž R . by Lemma 1.2. Thus, k␶ Ž R . s max k␶ Ž RA., k␶ Ž RrRA.4 s k␶ Ž RA. s k␶ Ž A., as required. PROPOSITION 2.3 Žcf. w15, Proposition 2.1x.. Let R be a ring with ␶-Krull dimension and Ž††.. Then the following statements hold. Ž1. Ž2.

If P g Min ␶ Ž R ., then E R Ž RrP . is ⌬-injecti¨ e. If C g F is ␣-␶-critical with ␣ s k␶ Ž R ., then E R Ž C . is ⌬-injecti¨ e.

Proof. Ž1. Let ␶ P s Ž TP , FP . be the torsion theory cogenerated by E R Ž RrP .. Since annihilators of nonempty subsets of E R Ž RrP . are ␶ Pclosed right ideals of R, the assertion will follow if we can prove that R is ␶ P-artinian. We first establish that R is ␶ P-semi-artinian by showing that any module 0 / X g FP contains a 0-␶ P-critical Žor ␶ P-simple. submodule. Now, FP : F ; so X is ␶-torsion-free; hence X contains a ␶-critical submodule C. By Ž††., there exists 0 / D F C such that rR Ž D . s Q g Spec ␶ Ž R .. As D is ␶ P -torsion-free, there exists 0 / f g Hom R Ž D, E R Ž RrP ... It follows from Ž RrP . R Fess E R Ž RrP . that f Ž D . l Ž RrP .R / 0; hence Q s rR Ž D . F rR Ž f Ž D . l Ž RrP . R . s P , so that Q s P, since P g Min ␶ Ž R .. By Lemma 2.2, RrP is k␶-homogeneous; so we get k␶ Ž RrP . s k␶ Ž f Ž D . l Ž RrP . R . F k␶ Ž D . F k␶ Ž RrQ . s k␶ Ž RrP . . Thus, k␶ Ž D . s k␶ Ž RrP .. Let 0 / D⬘ F D and suppose that DrD⬘ f TP . Then there exists 0 / ␾ g Hom R Ž DrD⬘, E R Ž RrP ... Since RrP is k␶-homogeneous and D is ␶-critical, it follows that k␶ Ž RrP . s k␶ Ž ␾ Ž DrD⬘ . l Ž RrP . R . F k␶ Ž DrD⬘ . - k␶ Ž D . s k␶ Ž RrP . , which is a contradiction. Therefore, DrD⬘ is ␶ P-torsion; so D is indeed a 0-␶ P-critical submodule of X. Thus, we have shown that R is ␶ P-semiartinian.

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Finally, R has ␶ P-Krull dimension, since R has ␶-Krull dimension and FP : F. Now apply w16, Theorem 3.2.9x to the lattice Sat ␶ PŽ R R . in order to conclude that k␶ PŽ R . s 0; that is, R is ␶ P-artinian. Ž2. By Ž††., there exists 0 / D F C with ␶ R Ž D . s P g Spec␶ Ž R .. Then,

␣ s k␶ Ž C . s k␶ Ž D . F k␶ Ž RrP . F k␶ Ž R . s ␣ ; so k␶ Ž RrP . s k␶ Ž R . s ␣ , and it follows from Lemma 1.2 that P g Min ␶ Ž R .. Therefore, E R Ž RrP . is ⌬-injective by Ž1.. We now finish the proof by showing that EŽ C . ¨ E R Ž RrP .. For this, let UrP be a ␶-critical submodule of RrP. Note that UrP is ␣-␶-critical, since RrP is k␶-homogeneous by Lemma 2.2. Since rR Ž D . s P - U, there exists 0 / d g D with dU / 0. Obviously, P F rR Ž d . l U s rU Ž d ., but this inclusion cannot be proper; for otherwise

␣ s k␶ Ž D . s k␶ Ž dU . s k␶ Ž UrrU Ž d . . - k␶ Ž UrP . s ␣ , which is a contradiction. Therefore, dU ( UrrU Ž d . s UrP; hence E Ž C . s E Ž D . s E Ž dU . ( E R Ž UrP . F E R Ž RrP . , which concludes the argument. The next lemma shows that Proposition 2.3 implies w6, Proposition 6x, which Beachy proved by using quotient categories. In fact, in view of Proposition 1.5, Beachy’s result is equivalent to Proposition 2.3Ž1.. LEMMA 2.4. MingŽ R ..

Let R be a ring with ␶-Krull dimension. Then Min ␶ Ž R . s

Proof. Let P g Min ␶ Ž R .. Then RrP g F ; so P g SpecgŽ R . by Lemma 1.1. We have to show that P g MingŽ R .. For this, let Q g SpecgŽ R . with Q : P. By w3, Lemma 11.23x, Q is either ␶-dense or ␶-closed in R. If Q were ␶-dense, it would follow that RrP g T ; hence RrP g T l F s 0, that is, P s R, which is a contradiction. Therefore, Q g Spec␶ Ž R .. Since Q : P and P g Min ␶ Ž R ., we deduce that Q s P, and consequently P g MingŽ R .. Conversely, if P g MingŽ R ., let Q g Spec␶ Ž R . with Q : P. Then Q g SpecgŽ R . by Lemma 1.1. Hence Q s P, which shows that P g Min ␶ Ž R .. Next, we present several other consequences of Proposition 2.3. All of them are relativizations of similar results established for rings with Žright. Krull dimension; their proofs are based on the corresponding ones in w15x. COROLLARY 2.5. Let R be a ring with ␶-Krull dimension, and assume that ␶ satisfies Ž†.. Then the following statements hold.

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Ž1. E R Ž RrN␶ Ž R .. is a ⌬-injecti¨ e module. Ž2. Any module MR with ␶-Krull dimension is noetherian with respect to the torsion theory cogenerated by E R Ž RrN␶ Ž R ... Proof. Ž1. Since RrN␶ Ž R .. is a semiprime right Goldie ring by Lemma 1.1, Min ␶ Ž R . is finite; so E R Ž RrN␶ Ž R .. ¨ [P g Min ␶ Ž R. E R Ž RrP .. Thus, E R Ž RrN␶ Ž R .. is ⌬-injective, since E R Ž RrP . is ⌬-injective for all P g Min ␶ Ž R . by Proposition 2.3Ž1.. For Ž2., reason as in w15, Corollaire 2.2x COROLLARY 2.6. Let R be a ring with ␶-Krull dimension, and assume that ␶ satisfies Ž†.. Then E R Ž M . is ⌬-injecti¨ e for any k␶-homogeneous module MR g F with k␶ Ž M . s k␶ Ž R .. Proof. By w6, Lemma 1x, M has finite uniform dimension; hence, using w16, Corollary 3.2.7x applied to the lattice Sat ␶ Ž M ., one deduces that M is an essential extension of a finite direct sum of ␣-␶-critical modules, where ␣ s k␶ Ž R .. Now apply Proposition 2.3Ž2.. COROLLARY 2.7. is right Goldie.

A ␶-torsion-free k␶-homogeneous ring R that satisfies Ž†.

Proof. By Corollary 2.6, E R Ž R . is ⌬-injective and hence also ⌺-injective by w3, Theorem 11.4x; so R R is a ⌺-module. Together with the fact that as a ␶-torsion-free module with ␶-Krull dimension R R has finite uniform dimension, this proves the assertion. The next result will be used frequently in the sequel. LEMMA 2.8. Let R be a ring with ␶-Krull dimension, and let P g Spec␶ Ž R .. Then there exists a ␶-torsion-free indecomposable injecti¨ e right R-module EP such that E R Ž RrP . ( EPn for some integer n G 1. Proof. By Lemma 1.1, RrP is right Goldie; so it is an essential extension of a finite direct sum of uniform right ideals UrP, i s 1, . . . , n. i Since the UrP are subisomorphic by w14, Corollary 3.3.3x, their injective i . are isomorphic. Set EP [ E R ŽUrP . and observe that hulls E R ŽUrP i i n n Ž . Ž . E R RrP ( [is1 E R UrP ( EP . i THEOREM 2.9. Let R be a ring with ␶-Krull dimension, and assume that ␶ satisfies Ž†.. Then the map w E x ¬ AssŽ E . induces a bijecti¨ e correspondence between the set of all isomorphism classes of ␶-torsion-free indecomposable ⌬-injecti¨ e right R-modules and the set Min ␶ Ž R .. Proof. Let E be a ␶-torsion-free indecomposable ⌬-injective right R-module with assassinator P g Spec␶ Ž R .. Since nonempty subsets of E are finitely annihilated, RrP ¨ E m and hence E R Ž RrP . ¨ E m for some

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positive integer m; so E R Ž RrP . is ⌬-injective. Furthermore, since E R Ž RrP . ( EPn for some integer n G 1 by Lemma 2.8, it follows that E ( EP . Next, we claim that P g Min ␶ Ž R .. To see this, let Q g Spec␶ Ž R . with Q : P. By Lemma 1.1, we have Q g SpecgŽ R .; hence Q is either ␶ P-closed or ␶ P-dense in R R by w3, Lemma 11.23x. But Q cannot be ␶ P-dense, for otherwise RrP g TP l FP s 0, which is a contradiction. Therefore, RrQ g FP ; so RrQ embeds in a direct product of copies of the cogenerator E R Ž RrP . of the ␶ P-torsion-free class FP . Since E R Ž RrP . has been shown to be a ⌬-module, RrQ ¨ ŽE R Ž RrP .. s for some integer s G 1. Therefore, s

 Q 4 s Ass Ž RrQ . : Ass Ž Ž E R Ž RrP . . . s  P 4 ; hence Q s P, which proves our claim. So far, we have shown that the map

␾ :  w E x N E ␶-torsion-free indecomposable ⌬-injective 4 ª Min ␶ Ž R . , where ␾ Žw E x. s AssŽ E ., is an injective map. Now, choose any P g Min ␶ Ž R .. Then E R Ž RrP . is ⌬-injective by Proposition 2.3. Since E R Ž RrP . ( EPn by Lemma 2.8, EP is ⌬-injective as well. As P s Ass Ž E R Ž RrP . . s Ass Ž EP . s ␾ Ž w EP x . , this proves that ␾ is also a surjective map. Note that any torsion theory ␶ on Mod-R such that R is ␶-noetherian satisfies condition Ž†. by w3, Lemma 9.1x. In particular, this holds if R is ␶-artinian in view of the Teply᎐Miller Theorem Žsee e.g. w3, Theorem 7.11x.. So, we have the next two results. COROLLARY 2.10 w3, Proposition 11.26x. The following hold for a right ␶-artinan ring R. Ž1. Spec␶ Ž R . s Min ␶ Ž R .. Ž2. The map w E x ¬ AssŽ E . is a bijection from SpecŽMod-R, ␶ . onto Spec␶ Ž R .; i.e., using the terminology of Section 4, the ring R has bijecti¨ e ␶-Gabriel correspondence. Proof. Ž1. Clearly, R is ␶-artinian if and only if k␶ Ž R . s 0. Hence, by using Lemma 1.2, we deduce that Spec␶ Ž R . s Min ␶ Ž R .. Ž2. Since R is ␶-artinian, every ␶-torsion-free right R-module is a ⌬-module. Now apply Theorem 2.9. The final result of this section is a relativization of w3, Corollary 11.27x.

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COROLLARY 2.11. If R is a right ␶-noetherian ring, then there exists a bijecti¨ e correspondence between the set Min ␶ Ž R . and the set of all isomorphism classes of ␶-torsion-free indecomposable ⌬-injecti¨ e right R-modules.

3. TAME MODULES The concept of a tame module was introduced by Jategaonkar w10x, but only for right noetherian rings. It has been extended to right ␶-noetherian rings in w11x, where two formally different Žbut still equivalent. forms were used. However, in the most general case, two distinct versions emerge, and the purpose of this section is to clarify their relationship. DEFINITION 3.1. The uniform right R-module U is Ži. P-tame if AssŽU . s P g SpecŽ R . and, for any 0 / u g lU Ž P ., the annihilator r Ž u. is not essential over P; that is, r Ž u.rP is not an essential right ideal of RrP; Žii. tame if it is P-tame for some prime ideal P. Note that a uniform P-primary module U is P-tame if and only if lU Ž P . is nonsingular as a right RrP-module. LEMMA 3.2. Let P be a prime ideal of R such that RrP is right Goldie. Then the following conditions are equi¨ alent for a uniform right R-module U. Ža. Žb. Žc. mand of

U is P-tame. EŽU . is P-tame. EŽU . ( EP Žs the unique indecomposable injecti¨ e direct sumE R Ž RrP ...

Proof. Ža. « Žb.. Since AssŽEŽU .. s AssŽU . s P, the module EŽU . is P-primary. Let 0 / x g l EŽU .Ž P . and assume that r Ž x .rP Fess RrP. Since lU Ž P . Fess EŽU ., there exists an element a g R _ r Ž x . such that 0 / xa g lU Ž P .. However, the right ideal B [  b g R N ab g r Ž x .4 is essential over P and B F r Ž xa.; so r Ž xa. is essential over P. Since we assume that U is P-tame, this gives a contradiction. Žb. « Ža.. Since lU Ž P . F l E ŽU .Ž P ., this is immediate. Žb. « Žc.. Choose 0 / u g l E ŽU .Ž P .. Since r Ž u. is not essential over P, there exists a right ideal A of R such that P - A and r Ž u. l A s P. Therefore, as right R-modules, A [ ArP s Ar Ž A l r Ž u . . ( uA F E Ž U . ; so EŽU . ( EŽ A. ( EP since EŽU . is indecomposable.

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Žc. « Ža.. Since EŽU . ( EP ( E R Ž VrP . for a uniform right ideal VrP of RrP and since the equivalence of Ža. and Žb. has already been established, it is sufficient to show that VrP is tame. Let 0 / ¨ q P g l V r P Ž P . s VrP, set A [ rR Ž ¨ q P . s  a g R N ¨ a g P 4 , and suppose that ArP Fess RrP. By Goldie’s Theorem, there exists an element c g A that is regular modulo P. Since ¨ c g P and ¨ f P, this is a contradiction. Remark 3.3. Note that Ža. m Žb. holds without the assumption that RrP is right Goldie. It also remains true that EŽU . is isomorphic to an indecomposable injective direct summand of E R Ž RrP ., although this need no longer be unique. DEFINITION 3.4. A right R-module M is Ži. weakly tame Ž weakly P-tame. if every uniform submodule of M is tame Ž P-tame.; Žii. tame Ž P-tame. if M is an essential extension of a direct sum of tame Ž P-tame. uniform modules; Žiii. X-tame for ⭋ / X : SpecŽ R ., if M is tame and AssŽ M . : X . Clearly, a module M is P-tame if and only if it is  P 4 -tame; it is SpecŽ R .-tame if and only if it is tame. Note also that a ␶-torsion-free right R-module is tame if and only if it is Spec␶ Ž R .-tame. Remark 3.5. A module without nonzero uniform submodules is vacuously weakly tame. The next lemma shows that ‘‘tame’’ implies ‘‘weakly tame’’ and that the two concepts are equivalent for ␶-torsion-free modules with ␶-Krull dimension, as those have finite uniform dimension. LEMMA 3.6. The following conditions are equi¨ alent for a nonzero right R-module M. Ža. M is tame. Žb. M is weakly tame and e¨ ery nonzero submodule of M contains a nonzero uniform module. Proof. Ža. « Žb.. Assume that M is tame and let N Fess M, where N s [i g I Ni is a direct sum of tame uniform submodules of M. Let U / 0 be a uniform submodule of M. Then U l N / 0; so let 0 / u g U l N. Now, u s n1 q ⭈⭈⭈ qn k with n i g Ni and  1, . . . , k 4 : I. Since the k sum of the Ni is direct, r Ž u. s Fis1 r Ž n i .. Since Rrr Ž u. ( uR F U is Ž . uniform, r u is an irreducible right ideal; so r Ž u. s r Ž n i . for some 1 F i F k. Therefore E Ž U . s E Ž uR . ( E Ž n i R . s E Ž Ni . .

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Since Ni is tame, EŽ Ni . and U are also tame by Lemma 3.2. Thus, M has been shown to be weakly tame. Now, let 0 / S F M and choose 0 / s g S l N. Then s is contained in a finite direct sum of uniform submodules; so sR has finite uniform dimension. Therefore, there exists a uniform module 0 / U F sR F S. Žb. « Ža.. Assume that M is weakly tame and that every nonzero submodule of M contains a uniform module. Let Ui N i g I 4 be the set of all uniform tame submodules of M. Call a subset J : I direct if Ý j g J Uj is direct. By Zorn’s Lemma, there exists a maximal direct subset Jmax : I. Let N s [j g J ma x Uj . Assume that N is not essential in M, and choose 0 / X F M such that N l X s 0. By hypothesis, X contains a uniform submodule U. Since M is assumed to be weakly tame, U is tame; so U s Ui for some i g I. But then Jmax j  i4 is a direct subset of I, which contradicts the maximality of Jmax . LEMMA 3.7. Let P be a prime ideal of the ring R, let M be a P-tame right R-module, and let 0 / m g l M Ž P .. Then r Ž m. is not essential o¨ er P. Proof. Let [i g I Ui be a direct sum of tame Žnaturally P-tame. uniform submodules of M such that [i g I Ui Fess M. Then [i g I lUiŽ P . Fess M; so there exists an element s g R such that 0 / ms g [i g I lUiŽ P .. Let ms s u1 q ⭈⭈⭈ qu n , where  1, . . . , n4 : I and u i g lUiŽ P . for 1 F i F n. Note that none of the annihilators r Ž u i . is essential over P; so the same is n true for r Ž ms . s Fis1 r Ž u i .. Assume that r Ž m. is essential over P. Then A s [  a g R N sa g r Ž m.4 is also essential over P. Since A s F r Ž ms ., this is a contradiction. It is easy to show that both the weakly tame and the X-tame right R-modules form classes that are closed under submodules, extensions, direct sums, and essential extensions. However, these classes are generally not closed under homomorphic images. LEMMA 3.8. Let R be a prime ring such that R R is a ⌬-module. Then R R is weakly 0-tame. Proof. Let U / 0 be a uniform right ideal of R Žif none exist, then the assertion is vacuously true.. Suppose that the singular submodule Z [  z g U N r Ž z . Fess R R 4 of UR is nonzero. As R R is assumed to be a ⌬-module, rR Ž Z . is a finite intersection of essential right ideals; hence rR Ž Z . / 0. Since Z ⭈ rR Ž Z . s 0 and R is prime, this is a contradiction. Therefore, UR is nonsingular; hence 0-tame. PROPOSITION 3.9. Let MR be a nonzero ⌬-module, and assume that for e¨ ery 0 / N F M there exists a submodule 0 / N⬘ F N such that r Ž N⬘. is a prime ideal. Then M is weakly tame.

RELATIVE GABRIEL CORRESPONDENCE

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Proof. Let U be a uniform submodule of M, and let 0 / U⬘ F U such k that r ŽU⬘. s P is a prime ideal. Since M is a ⌬-module, P s Fis1 r Ž ui . for finitely many elements u i g U⬘. Thus, RrP s R

k

k

is1

is1

F r Ž ui . ¨ [ ui R F U k .

Consequently, RrP is a ⌬-module with finite uniform dimension. By using Lemma 3.8, Lemma 3.6, and the fact that tameness is preserved under essential extensions, we conclude that E R Ž RrP . is P-tame; hence so is U, since EŽU . ¨ E R Ž RrP .. 4. BIJECTIVE ␶-GABRIEL CORRESPONDENCE The aim of this section is to show that some results from w5, 8, 11x, established for ␶-noetherian rings and rings with Krull dimension, hold more generally for any ring R with ␶-Krull dimension, where ␶ is a hereditary torsion theory on Mod-R satisfying the condition Ž†.; in particular, they hold whenever ␶ is an ideal invariant torsion theory. We begin by recalling several concepts that we shall use in the sequel. Let R be a ring with ␶-Krull dimension, let ER be a ␶-torsion-free indecomposable injective module, and let C be a ␶-critical submodule of E. Since E is uniform, k␶ Ž C . is independent of the choice of C. This invariant is therefore called the ␶-critical Krull dimension of E; it will be denoted by crk␶ Ž E .. The ring R is right ␶-bounded if every ␶-closed essential right ideal contains a nonzero ideal; it is right fully ␶-bounded if all prime factor rings of R are right ␶-bounded. The ring R is a ring with Žor which has. bijecti¨ e ␶-Gabriel correspondence if the assignment 䢇

䢇

䢇

Spec Ž Mod-R, ␶ . ª Spec␶ Ž R . ;

w E x ¬ Ass Ž E .

is a well-defined map, which is a bijection. In w5x and w11x, this condition has been called local bijecti¨ e Gabriel correspondence with respect to ␶ . Obviously, for a ring R to have bijective ␶-Gabriel correspondence, it is necessary for SpecŽMod-R, ␶ . and Spec␶ Ž R . to have the same cardinality. In particular, a ring fails to have bijective ␶-Gabriel correspondence if one of the two sets is empty, but the other is not. The examples below illustrate this.

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EXAMPLES 4.1. Ž1. Let R be a rank 1 nondiscrete valuation domain with its value group the additive group of real numbers, and let M be the Žunique. maximal ideal of R. Note that M s M 2 and that R has classical Krull dimension 1. Let ␶ be the torsion theory with torsion class T [  X R N XM s 04 . Let 0 / I - M and note that MrI f T since M s M 2 . Set KrI [ ␶ Ž MrI . - MrI and observe that no nonzero submodule of the ␶-torsion-free module MrK is annihilated by M; so M f AssŽ MrK .. Since Ž MrK . K s 0 and K / 0, the annihilator of a nonzero submodule of MrK cannot be 0; hence 0 f AssŽ MrK .. As 0 and M are the only primes of R, this means that AssŽ MrK . s ⭋; so ␶ does not satisfy Ž†.. Since the ideals of R are linearly ordered, MrK is uniform; so E R Ž MrK . is ␶-torsion-free indecomposable injective. The same is true for E R Ž R .; so cardŽSpecŽMod-R, ␶ .. G 2. However, Spec␶ Ž R . s  04 ; so cardŽSpec␶ Ž R .. s 1 and therefore R does not have bijective ␶-Gabriel correspondence. Now pass to the ring R⬘ s RrK and use the torsion theory ␶ ⬘ s ␶ K on Mod-Ž RrK .. Then M⬘ s MrK is the only prime ideal in R⬘. Since M⬘ is ␶ ⬘-dense in R⬘, it follows that Spec␶ ⬘Ž R⬘. s ⭋. However, SpecŽMod-R⬘, ␶ ⬘. X / ⭋ because MR⬘ is uniform, so that E R⬘Ž M⬘. is ␶ ⬘-torsion-free indecomposable injective. Ž2. Consider the ring R s F X, Y 4 of polynomials in two noncommuting indeterminates X and Y over a field F. Let ␶ s Ž T , F . be the Goldie Žor Lambek. torsion theory on Mod-R: T s  T N Hom R Ž TR , E Ž R R . . s 0 4 ;

F s  M N MR is nonsingular 4 .

Suppose that SpecŽMod-R, ␶ . / ⭋, and let E be a ␶-torsion-free indecomposable injective module. Then E is nonsingular; hence it contains a copy of a principal nonzero right ideal I of R. Since R R ( IR F E, this would imply that R R has uniform dimension 1. But R R has infinite uniform dimension. Thus, SpecŽMod-R, ␶ . s ⭋. However, Spec␶ Ž R . =  04 / ⭋. LEMMA 4.2. Let R be a ring, and let ␶ be a hereditary torsion theory on Mod-R. Suppose that e¨ ery nonzero ␶-torsion-free right R-module contains a uniform module / 0 and that R has bijecti¨ e ␶-Gabriel correspondence. Then R satisfies Ž†.. Proof. The requirement that AssŽ E . / ⭋ for a ␶-torsion-free indecomposable injective module E is implicit in the definition of bijective ␶-Gabriel correspondence. Therefore, if U is uniform, 0 / U F M g F , then ⭋ / AssŽEŽU .. s AssŽU . : AssŽ M .. In particular, rings with ␶-Krull dimension and bijective ␶-Gabriel correspondence satisfy Ž†.; this fact will be used without further mention from now on.

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661

The next two items extend results from w5x and w8x that were established for ␶-noetherian rings and rings with Krull dimension, respectively. LEMMA 4.3. Let R be a ring with ␶-Krull dimension, and let P g Spec␶ Ž R .. Then there exists a ␶-torsion-free indecomposable injecti¨ e module E, unique up to isomorphism, such that AssŽ E . s P and crk␶ Ž E . s k␶ Ž RrP .. Proof. Recall that RrP is k␶-homogeneous by Lemma 2.2; so the ␶-torsion-free indecomposable injective direct summand EP of E R Ž RrP . Žsee Lemma 2.8. is a module with the desired properties. For uniqueness up to isomorphism, let E be an arbitrary ␶-torsion-free indecomposable injective right R-module such that AssŽ E . s P and crk␶ Ž E . s k␶ Ž RrP ., and let C be a ␶-critical submodule of E. Since AssŽ C . s AssŽ E . s P, there exists a submodule 0 / D F C with rR Ž D . s P. Continue as in the proof of Proposition 2.3Ž2. to obtain an embedding EŽ C . ¨ E R Ž RrP .. It follows from Lemma 2.8 that E s EŽ C . ( EP . PROPOSITION 4.4. The following assertions are equi¨ alent for a ring R with ␶-Krull dimension. Ž1. R has bijecti¨ e ␶-Gabriel correspondence. Ž2. AssŽ E . / ⭋ and crk␶ Ž E . s k␶ Ž RrAssŽ E .. for e¨ ery ␶-torsionfree indecomposable injecti¨ e right R-module E. Proof. Apply Lemma 4.2 and Lemma 4.3. The following is a relative version of the corresponding result from w8x. Recall that a module is compressible if it can be embedded in each of its nonzero submodules. COROLLARY 4.5. Let R be a ring with ␶-Krull dimension and bijecti¨ e ␶-Gabriel correspondence. Then e¨ ery nonzero ␶-torsion-free right R-module contains a compressible module. Proof. By using Proposition 4.4, the proof is essentially the same as that of w8, Corollary 8.7x and is therefore omitted. We are now in a position to extend w11, Theorem 3.3x from ␶-noetherian rings to rings with ␶-Krull dimension. THEOREM 4.6. The following statements are equi¨ alent for a ring R with ␶-Krull dimension. Ži. R has bijecti¨ e ␶-Gabriel correspondence. Žii. Ža. ␶ is cogenerated by E R Ž[P g Spec Ž R. RrP .. ␶ Žb. Direct products of Spec␶ Ž R .-tame right R-modules are tame. Žiii. Any nonzero ␶-torsion-free right R-module is tame.

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Živ. For any ␶-torsion-free indecomposable injecti¨ e module E there exists a prime ideal P and a P-prime cyclic submodule eR F E with k␶ Ž eR . s k␶ Ž RrP .. Proof. Ži. « Žii.. To establish Žii.Ža., we need to show that each nonzero module M g F embeds in direct product of copies of ER

ž

[

/

RrP .

PgSpec ␶ Ž R .

Since every nonzero submodule of M has a nonzero Žcyclic. submodule with ␶-Krull dimension, there exists a family  Ci 4i g I of ␶-critical modules such that [i g I Ci Fess M. Hence M F EŽ[i g I Ci . F Ł i g I EŽ Ci .. So it is sufficient to show that EŽ Ci . ¨ E R Ž[P g Spec ␶ Ž R. RrP . for all i g I. Since R satisfies Ž†. by Lemma 4.2 and since Ci is uniform, AssŽ Ci . s Pi for some prime Pi g Spec␶ Ž R .. Then EŽ Ci . is ␶-torsion-free indecomposable injective with assassinator Pi ; so it follows from Ži. that EŽ Ci . ( EP i . Therefore, E Ž Ci . ( EP i ¨ E R Ž RrPi . F E R

ž

[

PgSpec ␶ Ž R .

/

RrP .

For Žii.Žb., first note that according to Definition 3.4 a Spec␶ Ž R .-tame module M is an essential extension of a direct sum of Spec␶ Ž R .-tame uniform modules U␣ . According to Definition 3.1, each U␣ contains a copy of a right ideal of RrP␣ , where P␣ s AssŽU␣ .. Since RrP␣ is ␶-torsion-free, EŽ M . is ␶-torsion-free Žas F is closed under direct sums and essential extensions.. Now let N s Ł␤ M␤ , where the M␤ are Spec␶ Ž R .-tame. Since each M␤ is ␶-torsion-free, then so is N Žas F is closed under direct products.. As in the proof of Žii.Ža., [i g I Ci Fess N, where each Ci is ␶-critical. By Ži., EŽ Ci . ( EP i for some Pi g Spec␶ Ž R .. By Lemma 3.2, each Ci is tame; hence the direct sum [i g I Ci is tame, and so is its essential extension N. Žii. « Žiii.. Let M / 0 be ␶-torsion-free. By Žii.Ža., M embeds in a direct product of modules of the form E R Ž RrP ., where P g Spec␶ Ž R .. Since each E R Ž RrP . is P-tame by Lemma 1.1, 2.8, and 3.2, such a product is tame by Žii.Žb.. Therefore, M is tame. Žiii. « Živ.. Let E be ␶-torsion-free injective indecomposable. By hypothesis, E is tame, say P-tame, where P g Spec␶ Ž R .. Let eR be a cyclic P-prime submodule of E; so r Ž e .rP is not an essential right ideal of RrP. Since RrP is k␶-homogeneous by Lemma 2.2, it follows that k␶ Ž eR . s k␶ Ž Rrr Ž e .. s k␶ Ž RrP .. Živ. « Ži.. Let E be a ␶-torsion-free indecomposable injective right R-module. If P and eR are as provided by Živ., then AssŽ E . s AssŽ eR . s

663

RELATIVE GABRIEL CORRESPONDENCE

P g Spec␶ Ž R .. By w16, Proposition 7.3.9x, r Ž e . is not essential over P; so Rrr Ž e . contains a submodule that is isomorphic to a uniform right ideal UrP of RrP. Therefore E s EŽ eR . ( EŽ Rrr Ž e .. ( E R ŽUrP . s EP . Clearly, this establishes the bijective ␶-Gabriel correspondence for R. COROLLARY 4.7. Suppose that R has ␶-Krull dimension and bijecti¨ e ␶-Gabriel correspondence. Then any homomorphic image RrI has bijecti¨ e ␶ I-Gabriel correspondence. Proof. Let M / 0 be a ␶ I-torsion-free right RrI-module. Then MR is ␶-torsion-free and hence tame by Theorem 4.6; so MR r I is also tame, and the assertion follows by applying Theorem 4.6 again. COROLLARY 4.8. Let R be a right fully ␶-bounded ring with ␶-Krull dimension and Ž†.. Then R has bijecti¨ e ␶-Gabriel correspondence. Proof. We show that statement Živ. of Theorem 4.6 holds. For this, let E be ␶-torsion-free indecomposable injective. By Ž†., AssŽ E . s P for some P g Spec␶ Ž R .. Choose 0 / e g l P Ž E . such that P s r Ž eR . F r Ž e .. Since r Ž e . is ␶-closed and RrP is ␶-bounded, r Ž e . cannot be essential over P; hence k␶ Ž eR . s k␶ Ž Rrr Ž e .. s k␶ Ž RrP . by Lemma 2.2. Remark 4.9. A ring with ␶-Krull dimension and bijective ␶-Gabriel correspondence is not necessarily right fully ␶-bounded, even if it is noetherian Žsee w11, Example 3.1x.. 5. ␶-KRULL DIMENSION AND CLASSICAL ␶-KRULL DIMENSION In w8, 12x, it was shown that cl. K dimŽ R . s K dimŽ R R . for every right FBN ring. In w13x, this has been relativized to non ␶-torsion right fully ␶-bounded rings with Krull dimension by defining the classical Krull dimension relative to a torsion theory ␶ as follows. Ž R . s ⭋, and for an ordinal ␣ ) y1 define Set Specy1 ␶ Spec␶␣ Ž R . s P g Spec␶ Ž R . N P F Q g Spec␶ Ž R . « Q g

½

D Spec␶␤ Ž R .

␤- ␣

5

.

If an ordinal ␣ with Spec␶␣ Ž R . s Spec␶ Ž R . exists, then the smallest such ordinal is called the classical ␶-Krull dimension of R; it is denoted by cl.K ␶ dimŽ R .. In w13, Theorem 3.2x, it was shown that if R R has ␶-Krull dimension, then R has classical ␶-Krull dimension as well, and cl.K ␶ dimŽ R . F k␶ Ž R R ..

664

ALBU, KRAUSE, AND TEPLY

In w11, Theorem 3.7x, the equality cl.K ␶ dimŽ R . s k␶ Ž R . has been established for a right ␶-noetherian ring R with bijective ␶-Gabriel correspondence. While we cannot fully extend this equality to rings with ␶-Krull dimension and bijective ␶-Gabriel correspondence, we are nevertheless able to show that the two dimensions differ by at most 1. The classical ␶-Krull dimension of a ring is a special instance of the more general concept of the classical Krull dimension of a poset Ž X, F. introduced in w1x, which we briefly describe below. For ordinals ␣ G y1, define subsets X␣ : X recursively as follows. Start by setting Xy1 s ⭋. Let ␣ G 0 and suppose that X␤ : X has been defined for each ␤ - ␣ . Set X␣ [ x g X N ᭙ y g X , x - y « y g

½

D X␤

␤- ␣

5

.

This construction leads to an ascending filtration Xy1 : X 0 : X 1 : ⭈⭈⭈ : X␣ : ⭈⭈⭈ , called the classical Krull filtration of X. Since X is a set, there exists a least ordinal ␥ Ž X . such that X␥ Ž X . s X␥ Ž X .q1. Obviously, X␥ Ž X . / ⭋ m X 0 / ⭋ m X has at least one maximal element. The poset X is said to have classical Krull dimension cl. K dimŽ X . s ␥ Ž X . if X s X␥ Ž X . . LEMMA 5.1 w1, Proposition 1.4x. The following assertions are equi¨ alent for a poset X. Ž1. Ž2.

X has classical Krull dimension. X is a noetherian poset.

Now consider a ring R and a hereditary torsion theory ␶ on Mod-R. Observe that the filtration  Spec␶␣ Ž R .4␣ Gy1 of SpecŽ R . defined at the beginning of this section is nothing else than the classical Krull filtration of Spec␶ Ž R .. PROPOSITION 5.2. Let R be a ring, and let r be a hereditary torsion theory on Mod-R. Then R has classical ␶-Krull dimension if and only if the poset ŽSpec␶ Ž R ., :. is noetherian. In particular, if R has ␶-Krull dimension dimension, then it has classical ␶-Krull dimension as well. Proof. Apply Lemma 5.1 and Proposition 1.3. LEMMA 5.3. Suppose that R has right ␶-Krull dimension and bijecti¨ e ␶-Gabriel correspondence. Let M be a right R-module with ␶-Krull dimension

665

RELATIVE GABRIEL CORRESPONDENCE

that is not ␶-torsion. Then there exists ⭋ / S : Spec␶ Ž R . such that r Ž M . : P for all P g S , and k␶ Ž M . s sup  k␶ Ž RrP . 4 . Pg S

In particular, k␶ Ž R . s sup k␶ Ž RrP . N P g Spec␶ Ž R .4 . Proof. Since k␶ Ž M . s k␶ Ž Mr␶ Ž M .., we may assume that M is ␶-torsion-free; so M is tame by Theorem 4.6. Consider the set X [  Ni 4i g I of ␶-closed submodules Ni that have the property claimed for M, and let Si denote the respective subsets of Spec␶ Ž R .. By Lemmas 1.1, 2.2, and 3.2, the ␶-closure of any P-prime submodule Žwhere P g AssŽ M . : Spec␶ Ž R .. is an element of X ; so X / ⭋. We proceed to show that M g X . Let  Nj 4j g J be an ascending chain of modules in X , let Nu s D j g J Nj, and define Su s Dj g J Sj . If P g Su , then P g Sj for some j g J; so P G r Ž Nj . G r Ž Nu .. By w16, Theorem 3.2.9x, k␶ Ž Nu . s k␶

žD /

Nj s sup  k r Ž Nj . 4 s sup sup  k␶ Ž RrP . 4

jgJ

jgJ

jgJ Pg Sj

s sup  k␶ Ž RrP . 4 ; Pg Su

so Nu g X . By Zorn’s Lemma, there exists a ␶-closed submodule Nm of M that is maximal in X . Suppose that Nm - M. Since MrNm is ␶-torsion-free and hence tame by Theorem 4.6, Lemma 3.2 provides a submodule UrNm F MrNm that is isomorphic to a uniform right ideal of RrQ for some Q g Spec␶ Ž R .. Let Nm⬘ s U; so k␶ Ž Nm⬘rNm . s k␶ ŽUrNm . s k␶ Ž RrQ . by Lemma 2.2. Set Sm⬘ s Sm j  Q4 . Observe that every prime in Sm⬘ contains r Ž Nm⬘ . and that k␶ Ž Nm⬘ . s max  k␶ Ž Nm⬘rNm . , k␶ Ž Nm . 4 s max k␶ Ž RrQ . , sup  k␶ Ž RrP . 4 s sup

½

Pg Sm

5

Pg Sm⬘

 k␶ Ž RrP . 4 .

Thus, Nm⬘ g X , which contradicts the maximality of Nm . THEOREM 5.4. Let R be a ring with right ␶-Krull dimension and bijecti¨ e ␶-Gabriel correspondence. Then cl.K ␶ dim Ž R . F k␶ Ž R R . F cl.K ␶ dim Ž R . q 1. Proof. Let k␶ Ž R . s ␣ . Since cl.K ␶ dimŽ R . F ␣ by w13, Theorem 3.2x, only the second inequality, k␶ Ž R R . F cl.K ␶ dimŽ R . q 1, has to be estab-

666

ALBU, KRAUSE, AND TEPLY

lished. By Corollary 4.7, homomorphic images of R also have bijective ␶-Gabriel correspondence; so we may assume that R is ␶-torsion-free and proceed by induction on ␣ . First note that we can reduce to the case in which R is prime. For, suppose that the desired equality has been established for all ␶-torsion-free prime factor rings of R. Then, by using Lemma 5.3, we have that k␶ Ž R . s sup  k␶ Ž RrP . N P g Spec␶ Ž R . 4 F sup  cl.K ␶ dim Ž RrP . q 1 N P g Spec␶ Ž R . 4 F cl.K ␶ dim Ž R . q 1. So let R be prime and ␶-torsion-free with k␶ Ž R . s ␣ . Since the inequality holds trivially for ␣ s y1, let ␣ ) y1 and assume that k␶ Ž RrP . F cl.K ␶ dimŽ RrP . q 1 has been established for every ␶-closed prime ideal P with k␶ Ž RrP . - ␣ . By w16, Proposition 7.3.9x, k␶ Ž R . s sup  k␶ Ž RrE . q 1 N E g Sat ␶ Ž R . , E Fess R R 4 . For each such E we have k␶ Ž RrE . s sup k␶ Ž RrP . N P g S 4 for some set S of ␶-closed prime ideals by Lemma 5.3. Now, for each P g S we have that k␶ Ž RrP . F k␶ Ž RrE . - ␣ ; so P ) 0. Hence cl.K ␶ dimŽ RrP . cl.K ␶ dimŽ R .. Therefore, by induction, k␶ Ž RrE . s sup  k␶ Ž RrP . N P g S 4 F sup  cl.K ␶ dim Ž RrP . q 1 N P g S 4 F cl.K ␶ dim Ž R . ; so k␶ Ž R . s sup k␶ Ž RrE . q 1 N E g Sat ␶ Ž R ., E Fess R R 4 F cl.K ␶ dimŽ R . q1. It is obvious that cl.K ␶ dimŽ R . s k␶ Ž R ., whenever the latter is a limit ordinal. This equality also holds if k␶ Ž R . - ⬁, which is a consequence of the following result. PROPOSITION 5.5. The following statements are equi¨ alent for a ring R with right ␶-Krull dimension and bijecti¨ e ␶-Gabriel correspondence. Ža. For e¨ ery limit ordinal y1 - ␭ - k␶ Ž R ., there exists a P g Spec␶ Ž R . with k␶ Ž RrP . s ␭. Žb. For e¨ ery ordinal y1 - ␤ - k␶ Ž R ., there exists a P g Spec␶ Ž R . with k␶ Ž RrP . s ␤ . Žc. k␶ Ž R . s cl.K ␶ dimŽ R . for e¨ ery homomorphic image R of R. Žd. k␶ Ž R . s cl.K ␶ dimŽ R . for e¨ ery prime homomorphic image R of R. Proof. Ža. « Žb.. Let ␤ - k␶ Ž R .. If ␤ is a limit ordinal, then there is nothing to prove; so let ␤ s ␥ q 1. Let M be a right R-module with k␶ Ž M . s ␤ . By Lemma 5.3, k␶ Ž M . s sup  k␶ Ž RrP . N P g S 4

for some subset S : Spec␶ Ž R . .

RELATIVE GABRIEL CORRESPONDENCE

667

Obviously, k␶ Ž RrP . F ␤ for all P g S . If k␶ Ž RrP . - ␤ for all P g S , then k␶ Ž RrP . F ␥ for all P g S , and it would follow that

␤ s k␶ Ž M . s sup  k␶ Ž RrP . N P g S 4 F ␥ - ␤ , which is a contradiction. Therefore, k␶ Ž RrP . s ␤ for some P g S . Žb. « Žc.. By Corollary 4.7, the conditions imposed on R are inherited by homomorphic images of R; so only k␶ Ž R . s cl.K ␶ dimŽ R . has to be established. As in the proof of Theorem 5.4, we may assume that R is prime ␶-torsion-free. Suppose that ␤ [ cl.K ␶ dimŽ R . - k␶ Ž R ., and take ␤ minimal with this property. Then k␶ Ž S . s cl.K ␶ dimŽ S . for every proper prime factor ring S of R. By Žb., there exists a ␶-closed prime ideal P ) 0 such that k␶ Ž RrP . s ␤ . By induction, cl.K ␶ dimŽ RrP . s ␤ . Since cl.K ␶ dimŽ RrP . - cl.K ␶ dimŽ R . s ␤ by the definition of classical ␶-Krull dimension, this is a contradiction. Žc. « Žd.. Trivial. Žd. « Ža.. Let ␭ be a limit ordinal, y1 - ␭ - k␶ Ž R .. By Lemma 5.3 and Žd.,

␭ - k␶ Ž R . s sup  k␶ Ž RrP . N P g Spec␶ Ž R . 4 s sup  cl.K ␶ dim Ž RrP . N P g Spec␶ Ž R . 4 ; so there exist ␶-closed prime ideals P with cl.K ␶ dimŽ RrP . G ␭, and Proposition 1.3 allows us to choose P maximal with respect to this property. For any ␶-closed prime ideal Q ) P, we then have that cl.K ␶ dimŽ RrQ . - ␭; hence cl.K ␶ dimŽ RrP . F ␭. Thus, cl.K ␶ dimŽ RrP . s ␭, and it follows from Žd. that k␶ Ž RrP . s ␭. COROLLARY 5.6.

Let R be a ring with right ␶-Krull dimension. If

Ži. R is right fully ␶-bounded and satisfies Ž†., or Žii. R is right ␶-noetherian with bijecti¨ e ␶-Gabriel correspondence, then cl.K ␶ dimŽ R . s k␶ Ž R . for e¨ ery homomorphic image R of R. Proof. Ži. By Corollary 4.8, R has bijective ␶-Gabriel correspondence. The result will follow from Proposition 5.5, if, for any limit ordinal ␭ - k␶ Ž R ., a ␶-closed prime ideal P can be found such that k␶ Ž RrP . s ␭. Since cl.K ␶ dimŽ R . F k␶ Ž R . F cl.K ␶ dimŽ R . q 1 by Theorem 5.4, ␭ F cl.K ␶ dimŽ R .. By Lemma 1.1, the semiprime ␶-torsion-free ring RrN␶ Ž R . is right Goldie; so R has only finitely many minimal ␶-closed prime ideals Pi , i s 1, . . . , n. Thus, we obtain

␭ F cl.K ␶ dim Ž R . s cl.K ␶ dim Ž RrN␶ Ž R . . F k␶ Ž RrN␶ Ž R . . s max  k␶ Ž RrPi . 4 , 1FiFn

668

ALBU, KRAUSE, AND TEPLY

whence M [  P g Spec␶ Ž R . N k␶ Ž RrP . G ␭4 / ⭋. Choose P0 to be maximal in M . We claim that k␶ Ž RrP0 . s ␭. For this, let ErP0 be a ␶-closed essential right ideal of RrP0 . By Lemma 5.3, k␶ Ž RrE . s sup k␶ Ž RrP . N P g S 4 for a set S of ␶-closed prime ideals P G r Ž RrE .. Since R is right fully ␶-bounded, D [ FP g S P G r Ž RrE . ) P0 ; whence k␶ Ž RrD . k␶ Ž RrP0 ., by Lemma 1.2. The ring RrD is semiprime right Goldie by Lemma 1.1; so there exists a minimal ␶-closed prime ideal QrD of RrD, such that k␶ Ž RrD . s k␶ Ž RrQ .. Since Q G D ) P0 , the maximality of P0 in M implies that k␶ Ž RrQ . - ␭. Therefore, k␶ Ž RrE . s sup  k␶ Ž RrP . N P g S 4 F k␶ Ž RrD . s k␶ Ž RrQ . - ␭ . Since ␭ is a limit ordinal, k␶ Ž RrE . q 1 - ␭, as well; hence

␭ F k␶ Ž RrP0 . s sup  k␶ Ž RrE . q 1 N E ␶-closed, ErP0 Fess RrP0 4 F ␭ , which establishes our claim. Žb. If R is right ␶-noetherian, then R has right ␶-Krull dimension and satisfies Ž†.. Given any ␤ F k␶ Ž R ., there exists a subfactor M of R with k␶ Ž M . s ␤ by w14, Proposition 6.1.7x. As a ␶-noetherian module, M has a ␤-␶-critical factor C s MrN. By Theorem 4.6, C is P-tame for some P g Spec␶ Ž R .; so C contains a submodule U / 0 that is isomorphic to a uniform right ideal of RrP. Consequently, k␶ Ž RrP . s k␶ ŽU . s k␶ Ž C . s ␤ ; hence the result follows from Proposition 5.5. 6. ⌬-MODULES OVER RINGS WITH ␶-KRULL DIMENSION The aim of this section is to investigate ⌬-modules over rings with ␶-Krull dimension. We show that a series of results in w11x involving this concept can be extended from ␶-noetherian rings to rings with ␶-Krull dimension that satisfy Ž†.. The facts below will be used in what follows without further comment. Let M be a P-tame right R-module, where P g Spec␶ Ž R .. M is ␶-torsion-free, because M is an essential extension of a direct sum of uniform P-tame right R-modules, each of which embeds in E R Ž RrP . Žsee Lemma 3.2 and Remark 3.3.. l M Ž P . Fess M, since [i g I Ui Fess M for P-tame uniform submodules Ui , since lUiŽ P . F ess Ui for each i g I, and since [i g I lUiŽ P . F l M Ž P .. If R has ␶-Krull dimension, then k␶ Ž N . s k␶ Ž RrP . for any submodule 0 / N F l M Ž P . with ␶-Krull dimension. This is easily deduced from Lemma 3.7 and the fact that RrP is k␶-homogeneous by Lemma 2.2. 䢇

䢇

䢇

669

RELATIVE GABRIEL CORRESPONDENCE

LEMMA 6.1. Let P be a prime ideal of the ring R, and let M be a P-tame right R-module. If M is a ⌬-module, then M I is P-tame for any index set I / ⭋. Proof. By Lemma 3.6, in order to establish that M I is P-tame for an arbitrary set I, we have to prove Ži. that every uniform submodule of M I is P-tame and Žii. that every nonzero submodule of M I contains a uniform submodule. Both Ži. and Žii. will follow if we can show that every nonzero cyclic submodule mR of M I is P-tame. For this, let m s Ž m i . i g I g M I, where m i g M for all i g I. Now, k

r Ž m . s r Ž Ž m i . igI . s

F r Ž mi . s r Ž  mi N i g I 4 . s F r Ž mi . , igI

is1

where  1, . . . , k 4 : I, since M is assumed to be a ⌬-module. Thus, mR ( Rrr Ž m . s R

k

k

is1

is1

F r Ž mi . ¨ [ mi R F M k .

Since M is P-tame, M k is P-tame; so mR is P-tame as well. The next result shows that the converse holds if R has ␶-Krull dimension and P is a ␶-closed prime ideal. PROPOSITION 6.2. Let R be a ring with ␶-Krull dimension, let P g Spec␶ Ž R ., and let M be a P-tame right R-module. Then the following statements are equi¨ alent. Ža. Žb.

M I is P-tame for any index set I / ⭋. M is a ⌬-module.

Proof. Ža. « Žb.. Let S be a nonempty subset of M. We proceed to show that S is finitely annihilated. Since Rrr Ž S . ¨ M S and M S is P-tame by hypothesis, Rrr Ž S . is also P-tame. Let Lrr Ž S . [ l R r r Ž S . Ž P . Fess Rrr Ž S . . Now let s1 g S. If A1 [ L l r Ž s 1 . s r Ž S . , then Lrr Ž S . Fess Rrr Ž S . implies that r Ž s1 . s r Ž S ., and it is done. Otherwise there exists an element s2 g S such that A1 g r Ž s2 .. Set A 2 [ L l r Ž s 1 . l r Ž s 2 . - A1 .

670

ALBU, KRAUSE, AND TEPLY

If A 2 s r Ž S ., it is done; otherwise continue in this fashion to obtain a strictly descending chain of right ideals L G A1 ) A 2 ) ⭈⭈⭈ ) A i ) A iq1 ) ⭈⭈⭈ G r Ž S . , where A i [ L l r Ž s1 , s2 , . . . , si .. Note that A irA iq1 s A ir Ž A i l r Ž siq1 . . ( siq1 A i and that siq1 A i P F siq1 LP F siq1 r Ž S . s 0; so siq1 A i F l M Ž P .. Each siq1 A i has ␶-Krull dimension; so the third of the facts listed before Lemma 6.1 implies that k␶ Ž A irA iq1 . s k␶ Ž siq1 A i . s k␶ Ž RrP . s k␶ Ž Lrr Ž S . . whenever A i ) A iq1. Thus, it follows from the definition of the ␶-Krull dimension that the chain of the A i ’s has only finitely many proper inclusions; so A i s r Ž S . for some i; whence r Ž S . s r Ž s1 , s2 , . . . , si . since Lrr Ž S . Fess Rrr Ž S .. Therefore, every nonempty subset of M is finitely annihilated, which implies that M is a ⌬-module. Žb. « Ža.. This is the assertion of Lemma 6.1. In w11x, it has been shown that for a right ␶-noetherian ring R the condition Ža. of the preceding result is satisfied if and only if P satisfies the so-called right ␶-restricted strong second layer condition. This condition was formulated like the strong second layer condition for noetherian rings introduced by Jategaonkar in w10x, except that the class of modules was ‘‘restricted’’ from finitely generated P-primary ones to ␶-noetherian P-tame ones. Here we modify this condition once more Žin two different ways. and establish analogues of w11, Proposition 5.1 and Theorem 5.2x for a ring R with right ␶-Krull dimension that satisfies Ž†.. DEFINITION 6.3. Let R be a ring with right ␶-Krull dimension. A prime ideal P g Spec␶ Ž R . satisfies the right restricted Žresp. ␶-restricted . strong second layer condition if, given a right R-module M that is P-tame Žresp. P-tame with ␶-Krull dimension. and a prime annihilator r Ž M . s Q g Spec␶ Ž R ., then Q s P. A nonempty subset S of Spec␶ Ž R . satisfies the right restricted Ž␶-restricted . strong second layer condition if each P g S satisfies the respective condition. PROPOSITION 6.4. The following statements are equi¨ alent for P g Spec␶ Ž R ., where R is a ring with ␶-Krull dimension that satisfies Ž†..

671

RELATIVE GABRIEL CORRESPONDENCE

Ža. P satisfies the right restricted Ž␶-restricted . strong second layer condition. Žb. M I is P-tame for any index set I / ⭋ and any right R-module M that is P-tame Ž P-tame with ␶-Krull dimension.. Žc. E¨ ery right R-module that is P-tame Ž P-tame with ␶-Krull dimension. is a ⌬-module. Žd. E¨ ery right R-module that is P-tame Ž P-tame with ␶-Krull dimension. is finitely annihilated. Proof. Ža. « Žb.. Let MR be P-tame Žresp. P-tame with ␶-Krull dimension.. Note that M is ␶-torsion-free; so M I is ␶-torsion-free. Let 0 / m g M I. Then mR ( Rrr Ž m. is a torsion-free right R-module with ␶-Krull dimension; so it has finite uniform dimension by w16, Corollary 7.3.5x. Therefore, every nonzero submodule of M I contains a uniform submodule. Now, let U be a uniform submodule of M I. Since Ž†. is assumed, AssŽU . / ⭋; so let Q s r Ž uR . g AssŽU ., where u s Ž u i . i g I g U, u i g M for all i g I. Then Q s r Ž Ž u i . igI R . s

F r Ž ui R . s r Ý ui R

ž

igI

igI

/.

As a submodule of M, Ý i g I u i R is P-tame Žresp. P-tame with ␶-Krull dimension.; so Q s P, since P satisfies the right restricted Žresp. ␶-restricted. strong second layer condition. To see that U is P-tame, let u s Ž u i . i g I g lU Ž P .. Since M is P-tame, none of the right ideals r Ž u i .rP is essential in RrP by Lemma 3.7; so r Ž u. s Fi g I r Ž u i . is not essential over P either. Žb. « Žc.. This follows from Proposition 6.2. Žc. « Žd.. Trivial. Žd. « Ža.. Let MR be P-tame Žresp. P-tame with ␶-Krull dimension. and assume that r Ž M . s Q is a prime ideal. Since M is finitely annihilated, k k Q s r Ž M . s Fis1 r Ž m i ., m i g M; so RrQ ¨ [is1 Rrr Ž m i ., and hence k

 Q 4 s Ass Ž RrQ . : Ass

ž[ is1

k

Rrr Ž m i . s Ass

/

ž[ / is1

m i R : Ass Ž M k .

s Ass Ž M . s  P 4 , and Q s P is now trivial. Remark 6.5. Proposition 2.3Ž1. is a corollary of the preceding result. For, if P g Min ␶ Ž R ., then P clearly satisfies the right restricted strong second layer condition. Since E R Ž RrP . is P-tame, it is therefore a ⌬module.

672

ALBU, KRAUSE, AND TEPLY

For the proof of our final result, note that if R has ␶-Krull dimension and Mfin is a finitely generated right R-module, then Mfin has ␶-Krull dimension and k␶ Ž Mfin . F k␶ Ž R .; moreover, if E is a ␶-closed right ideal of R, then RrE has ␶-Krull dimension and a finite assassinator. THEOREM 6.6. Let R be a ring with ␶-Krull dimension that satisfies Ž†., where ␶ is a hereditary torsion theory on Mod-R. Then the following statements are equi¨ alent. Ž1. Ža. R has bijecti¨ e ␶-Gabriel correspondence. Žb. Spec␶ Ž R . satisfies the right restricted Ž resp. ␶-restricted . strong second layer condition. Ž2. E¨ ery ␶-torsion-free right R-module with finite assassinator Ž resp. with ␶-Krull dimension. is a ⌬-module. Ž3. E¨ ery ␶-torsion-free right R-module M with finite assassinator Ž resp. with ␶-Krull dimension. has a finitely generated submodule Mfin such that k␶ Ž Mfin . s k␶ Ž Rrr Ž M ... Ž4. E¨ ery ␶-torsion-free right R-module M with finite assassinator Ž resp. with ␶-Krull dimension. has a finitely generated submodule Mfin with k␶ Ž Mfin . s cl.K ␶ dimŽ Rrr Ž M ... Ž5. Ža. R is right fully ␶-bounded. Žb. Spec␶ Ž R . satisfies the right restricted Ž resp. ␶-restricted . strong second layer condition. Proof. Ž1. « Ž2.. Let M be a ␶-torsion-free right R-module. The hypothesis Ž1.Ža. implies that M is tame by Theorem 4.6. First, assume that Spec␶ Ž R . satisfies the right restricted second layer condition and that M has finite assassinator, say AssŽ M . s  P1 , . . . , Pn4 : Spec␶ Ž R .. Now, M is an essential extension of a direct sum of uniform modules U␣ , each of which has nonempty assassinator AssŽU␣ . : AssŽ M .. For i s 1, . . . , n, let Ni be the direct sum of the U␣ with AssŽU␣ . s Pi . n Clearly, M ¨ [is1 EŽ Ni .. Since each E Ž Ni . is Pi-tame, it is a ⌬-module by Proposition 6.4; hence so is M. Next, assume that Spec␶ Ž R . satisfies the right ␶-restricted second layer condition and that M has ␶-Krull dimension. Then M has finite uniform dimension, say k; so there exist ␶-closed submodules N1 , . . . , Nk of M such that 0 s F1 F i F k Ni is an irredundant meet and each Ui s MrNi is uniform Žsee e.g., w16, Theorem 1.5.10x.. Obviously, each Ui has ␶-Krull dimension, and since it is P-tame for some P g Spec␶ Ž R . by Theorem 4.6 n it is therefore a ⌬-module by Proposition 6.4. As M ¨ [is1 Ui , the same holds for M. Ž2. « Ž3.. Let MR be ␶-torsion-free with finite assassinator Žresp. with ␶-Krull dimension.. Since M is assumed to be a ⌬-module, r Ž M . s

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n Fis1 r Ž m i ., m i g M. Let Mfin s Ý nis1 m i R, and note that r Ž M . s r Ž Mfin . and that Mfin has ␶-Krull dimension. Since

n

Rrr Ž M . s Rrr Ž Mfin . s R

F r Ž mi . ¨ is1

n

[m R F Ž M is1

i

n

fin

. ,

it follows that k␶ Ž Rrr Ž M .. s k␶ Ž Rrr Ž Mfin .. F k␶ Ž Mfin .. As k␶ Ž Mfin . F k␶ Ž Rrr Ž M .. holds in general, the desired equality follows. Ž3. « Ž4.. Obviously, it suffices to establish k ␶ Ž Rrr Ž M .. s cl.K ␶ dimŽ Rrr Ž M ... This will follow from Corollary 5.6Ži., provided that we can show that R is right fully ␶-bounded. For this, let P be a prime ideal, and let E be a ␶-closed right ideal that is essential over P. If P is not ␶-closed, then E G P ) P. Since P is a two-sided ideal, RrP is therefore right ␶-bounded. If, on the other hand, P is ␶-closed, then k␶ Ž RrE . k␶ Ž RrP ., since RrP is right Goldie by Lemma 1.1. Since our hypothesis implies that k␶ Ž RrE . G k␶ Ž FrE . s k␶ Ž Rrr Ž RrE .. for some finitely generated right ideal F ) E, it follows that r Ž RrE . ) P. Ž4. « Ž5.. In order to establish Ž5.Ža., let P be a prime ideal of R and let E be a ␶-closed right ideal that is essential over P. As in the previous paragraph, E G P ) P, if P is not ␶-closed. So let P be ␶-closed. Again, observe that k ␶ Ž RrP . ) k ␶ Ž RrE .. By hypothesis, k ␶ Ž ArP . s cl.K ␶ dimŽ RrrR Ž RrP .. for some finitely generated right ideal A ) P. Since rR Ž ArP . s P and since k␶ Ž ArP . s k␶ Ž RrP . by Lemma 2.2, it follows that k␶ Ž RrP . s k␶ Ž ArP . s cl.K ␶ dim Ž RrrR Ž ArP . . s cl.K ␶ dim Ž RrP . . Our hypothesis also implies that k␶ Ž FrE . s cl.K ␶ dimŽ Rrr Ž RrE .. for some finitely generated right ideal F ) E. Suppose that r Ž RrE . s P. Then k␶ Ž RrP . ) k␶ Ž RrE . G k␶ Ž FrE . s cl.K ␶ dim Ž Rrr Ž RrE . . s cl.K ␶ dim Ž RrP . s k␶ Ž RrP . , which is an obvious contradiction. Therefore, r Ž RrE . ) P. For Ž5.Žb., let P g Spec␶ Ž R ., let MR be P-tame Žresp. P-tame with ␶-Krull dimension., and assume that r Ž M . s Q g Spec␶ Ž R .. We proceed to show that Q s P in either case. By Ž4., k␶ Ž Mfin . s cl.K ␶ dimŽ Rrr Ž M .. for some finitely generated submodule Mfin F M. Furthermore, k␶ Ž RrQ . s cl.K ␶ dimŽ RrQ . by Ž5.Ža. and Corollary 5.6. Thus, k␶ Ž Mfin . s cl.K ␶ dim Ž Rrr Ž M . . s cl.K ␶ dim Ž RrQ . s k␶ Ž RrQ . .

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We claim that r Ž Mfin . s Q. Indeed, Q s r Ž M . F r Ž Mfin .. Assume that this inclusion is strict. Then, by Lemma 1.2, we would have that k␶ Ž Mfin . F k␶ Ž Rrr Ž Mfin .. - k␶ Ž RrQ ., which contradicts the equality above. Since Q is a prime ideal, Q s r Ž mR. for some element 0 / m g Mfin . Note that r Ž m. is ␶-closed. Since R is right fully ␶-bounded, r Ž m. cannot be essential over Q, so that r Ž m. l A s Q for some right ideal A ) Q. Thus,

 Q 4 s Ass Ž ArQ . s Ass Ž Ar Ž r Ž m . l A . . s Ass Ž mA . : Ass Ž M . s  P 4 ; so Q s P. Ž5. « Ž1.. By Corollary 4.8, Ž1.Ža. follows from Ž5.Ža..

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