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When R is a testing module for projectivity?
Accepted Manuscript When R is a testing module for projectivity?
Hayder Abd Albaqir Alhilali, Yasser Ibrahim, Gena Puninski, Mohamed Yousif
PII: DOI: Reference:
S0021-8693(17)30250-8 http://dx.doi.org/10.1016/j.jalgebra.2017.04.010 YJABR 16197
To appear in:
Journal of Algebra
Received date:
7 January 2016
Please cite this article in press as: H.A.A. Alhilali et al., When R is a testing module for projectivity?, J. Algebra (2017), http://dx.doi.org/10.1016/j.jalgebra.2017.04.010
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
WHEN R IS A TESTING MODULE FOR PROJECTIVITY? HAYDER ABD ALBAQIR ALHILALI, YASSER IBRAHIM, GENA PUNINSKI, AND MOHAMED YOUSIF Abstract. A right R-module is called R-projective if it is projective relative to the right R-module RR . A ring R is called right testing for projectivity if every right R-projective module is projective. Every right perfect ring is right testing and there are examples of local rings that are not right testing. In this paper an attempt is made to understand the class of right testing rings and several examples are provided to highlight the set theoretic obstacles in characterizing such rings.
1. Introduction If M and N are right R-modules, then M is called N -projective (projective relative to N ) if every R-homomorphism from M into an image of N can be lifted to an R-homomorphism from M into N . M is called R-projective, if it is projective relative to the right R-module RR . The module M is called projective, if M is N -projective, for every right R-module N . Dually, we can define, N -injective, R-injective, and injective modules. Given any module M . An R-monomorphism σ : M → E is called an injective envelope (injective hull) of M if E is injective and σ(M ) ⊆ess E, i.e. K ∩ σ(M ) = 0, whenever 0 = K ⊆ E. Dually, an R-epimorphism φ : P → M is called a projective cover of the right R-module M , if P is projective and ker φ is small in P , i.e. L + ker φ = M for any proper submodule L of M . A result of Eckmann and Schopf [6] asserts that every right R-module M can be embedded in an injective envelope (hull) of M . In dualizing this result, Bass [2] has shown that every (finitely generated) right R-module has a projective cover if and only if R is a right (semi) perfect ring. On the other hand, a result of Baer [3] known by the Baer Criterion asserts that a right R-module M is injective if and only if it is injective relative to RR . In general, the dual to the Baer Criterion is not true as there are examples of R-projective modules that are not projective. While it is not difficult to see that every finitely generated R-projective module is projective (see Lemma 2.1 below), the situation is quite different if the module is not finitely generated. We will say that a ring R is right testing for projectivity if every right Rprojective module is projective. In his remarks at the end of chapter 22, [8], Faith left the characterization of right testing rings as an open problem. In [16], Sandomierski proved that every right perfect ring is a right testing ring. In [12], Ketkar and Vanaja extended the work of Sandomierski by showing that if R is a semiperfect ring then every R-projective module with small radical is projective. However, if R = Z(p) is the localization of the ring of integers with respect to the prime ideal 2010 Mathematics Subject Classification. Primary 16D40, 16D50, 16D60; Secondary 16L30, 16L60, 16P20, 16P40, 16P60. Key words and phrases. Projective modules, R-projective Modules, Baer Criterion, Bass rings. 1
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pZ, then R is a commutative noetherian valuation domain. In particular R is local and every nonzero ideal of R has the form pn R for some n. If M = R/pR is the unique simple R-module and E = E(M ) is the injective envelope of M , then clearly Hom(E, R/I) = 0 for every ideal I of R. Therefore E is R-projective but not projective. This shows that semiperfect (indeed local) rings need not be right testing. However, a result of Hamsher [9] asserts that if R is a commutative noetherian ring then R is a testing ring iff R is artinian, and in this paper we will show below that if R is a semilocal right noetherian ring, then R is right testing iff R is right artinian. Most recently, in [5, Theorem 3] the authors claimed that if R is a right hereditary right noetherian V -domain that is not artinian, then R is a right testing ring. Unfortunately the proof of this result contains a gap, and we will show below that such a result can not be carried out in the standard Zermelo-Frenkel axiomatic set theory with the axiom of choice, ZF C. Indeed, the work of Trlifaj in [18] shows that the characterization of right testing rings may depend on some set-theoretic axioms which are independent of ZF C. Throughout, rings R are associative with unity and modules are unitary right R-modules, unless specified otherwise. For a module M , we use rad(M ) and E(M ) to denote the Jacobson radical and the injective hull of M, respectively. If M = R, we write J(R) = rad(R). We write N ⊆ M if N is a submodule of M , N ⊆ess M if N is an essential submodule of M , and N ⊆⊕ M if N is a direct summand of M . We say that a submodule N of M is a small submodule and write N M if M = N + L for any proper submodule L M . We write M (I) for a direct sum of I-copies of M , M I for a direct product of I-copies of M, and Mod-R for the category of right R-modules. 2. On right hereditary right noetherian V -domains We start with the following observation. Lemma 2.1. Every finitely generated R-projective module M is projective. η
Proof. Let R(n) −→ M → 0 be the natural R-epimorphism with n 1. By [1, Theorem 16.12], since M is R-projective, M is R(n) -projective and the map η splits. Therefore M is projective. Definition 2.2. A ring R is called right testing for projectivity (or just right testing) if every right R-projective module is projective. As we have seen in the proof of Lemma 2.1 above, a right R-module M is Rprojective if and only if M is R(n) -projective, for every n 1. Now the next result is a straightforward application of [1, Proposition 21.6]. Proposition 2.3. If R and S are Morita equivalent rings, then R is right testing if and only if S is right testing. While it will be interesting to look for other consistent set theoretic universes where right testing but not right perfect rings exist, we will limit our work below within the well-established ZF C with some additional mild set-theoretic axioms that are consistent with ZF C. In this direction, we will put to use the following theory developed by Trlifaj in [18]: A class of (right) R-modules M is said to be ptesting if whenever N is a right R-module such that Ext(N, M ) = 0 for every right R-module M ∈ M then N is projective. For instance, the class of all R-modules is
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p-testing. If k is a cardinal of cofinality ℵ0 , Trlifaj showed in [18, Lemma 2.4] how to use Shelah’s uniformization principle U Pk (which is consistent with ZF C + GCH) to disprove the existence of proper p-testing classes by establishing the next result. For a definition of Shelah’s uniformization principle U Pk , where k is a cardinal of cardinality ℵ0 , see [18, Page 1526]. Lemma 2.4. Suppose that R is not a right perfect ring and let k be an uncountable cardinal of cofinality ℵ0 . If U Pk holds, then there exists a non-projective right Rmodule N such that Ext(N, M ) = 0 for every right R-module M of cardinality < k. The non-projective module N in the proof of Lemma 2.4 above is a flat module of projective dimension 1. Furthermore if k |R| + ℵ0 , then N is k-generated. In the next proposition we provide a sufficient condition for an R-module to be R-projective. Proposition 2.5. If M is a right R-module such that Ext(M, I) = 0 for every non-zero proper right ideal I of R, then M is R-projective. In particular, if R is a right testing ring then the class of proper right ideals of R is p-testing. Proof. By applying Hom(M, −) to the short exact sequence 0 → I → R → R/I → 0, we obtain the following exact sequence: 0 → Hom(M, I) → Hom(M, R) → Hom(M, R/I) → Ext(M, I) → Ext(M, R) → ... If Ext(M, I) = 0 for every proper right ideal I of R, it follows that M is Rprojective. The last assertion is clear if R is right testing. In the next result we establish a converse to Proposition 2.5 in the case of selfinjective rings. Proposition 2.6. Let R be a right self-injective ring and M a right R-module. Then M is R-projective if and only if Ext(M, I) = 0 for every right ideal I of R. Proof. Consider the long exact sequence in the proof of Proposition 2.5 above. Since R is right self-injective, Ext(M, R) = 0. Then the map Hom(M, R) → Hom(M, R/I) is onto if and only if Ext(M, I) = 0 for every right ideal I of R. In the next theorem we show that right testing rings are right perfect if one assumes Shelah’s uniformization principle U Pk . Theorem 2.7. Let R be a ring of cardinality λ and let k = (λ+ )ω . If U Pk is assumed, then R is right testing if and only if R is right perfect. Proof. It sufficies to check only the forward implication. So assume that R is right testing. If R is not right perfect, then by Lemma 2.4, there exists a non-projective right R-module N such that Ext(N, I) = 0 for every proper right ideal I of R. Now, by Proposition 2.5, we infer that N is projective, a contradiction. Remark 2.8. In [5, Theorem 3] the authors claimed that if R is a right hereditary right noetherian V -domain that is not artinian, then R is a right testing ring, where a ring R is called a right V -ring (after Villamayor) if every simple right R-module is injective. Unfortunately there is a gap in the second pragraph of the proof of this result. However, since such a ring is not right perfect, assuming (sufficiently large) U Pk , we infer from Theorem 2.7 that the ring R cannot be right testing. Thus the
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proof of [5, Theorem 3] cannot be carried out in ZF C. Of course this does not prevent [5, Theorem 3] to be true in other ZF C extensions. Indeed, we have no answer to the following general question: Is it consistent with ZF C that there exists a right testing ring that is not right perfect? The following is a specific example of a hereditary noetherian V -domain, due to Cozzens [4] (see also [11]) that is neither left nor right testing ring if one assumes Shelah’s uniformization principle U Pk . Example 2.9. Let K be a universal differential field with a derivation δ and let R = K [y; δ] denote the ring of differential polynomials in the indeterminate y with coefficients in K, where addition is defined in the obvious way and multiplication defined by yk = ky + δ(k) for all k ∈ K. Then R is a right (and left) principal ideal domain. R is also a right (and left) P CI-ring; i.e. R/I is injective whenever I is a nonzero right (left) ideal of R. In particular, simple right (and left) R-modules are injective (i.e. R is right and left V -ring). Observe that the trivial module K is the unique (up to isomorphism) simple module of R and R/I ∼ = K n , for some n, whenever I is a nonzero right (left) ideal of R. If one assumes K to be countable, then the ring R = K [y; δ] is also countable and is neither right nor left perfect ring. If Shelah’s uniformization principle U Pk for k = (ℵ1 )ω is assumed, then R is neither left nor right testing. Recall that a ring R is called a right B-ring (the B stands for Bass) if every nonzero right R-module contains a maximal submodule. It is known that, R is a right B-ring if and only if J(R) is left T -nilpotent and R/J(R) is a right B-ring, see for example [17, Lemma 1]. Furthermore, Koifman, [13, Theorem 3], characterized commutative B-rings as those rings whose J(R) is T -nilpotent and R/J(R) is a von Neumann regular ring. For Noetherian rings we have the following. Lemma 2.10. Every right noetherian right testing ring R is a right B-ring. In particular J(R) is nilpotent. Proof. Assume to the contrary that there exists a nonzero right R-module M with no maximal submodules. We claim that Hom(M, R/I) = 0 for every right ideal I of R. Otherwise, let f : M → R/I be a nonzero R-homomorphism. Since f (M ) is a submodule of the right noetherian R-module R/I, it contains a maximal submodule N . But then f −1 (N ) is a maximal submodule of M , a contradiction. Thus M is R-projective, and hence projective since R is right testing. Since every projective module contains a maximal submodule, we obtain another contradiction. The last assertion follows from the above remarks and Levitzky’s theorem which states that if a ring R is right neoetherian then every one-sided nil ideal is nilpotent (see [1, Theorem 15.22]). A result of Hamsher [9] asserts that if R is a commutative noetherian ring then R is a testing ring if and only if R is artinian. In the next proposition we obtain a noncommutative version of Hamsher’s result for semilocal right noetherian rings. Proposition 2.11. Let R be a semilocal right noetherian ring. Then R is right testing if and only if R is right artinian. Proof. We only need to prove the forward implication (⇒). By Lemma 2.10, since R is right noetherian, J(R) is nilpotent. Thus R is a semiprimary ring, and hence right artinian.
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Since every right noetherian right testing ring is a right B-ring and every commutative B-ring R has R/J(R) regular, Hamsher’s result is now an immediate consequence of Proposition 2.11. Remark 2.12. (1) The following seems to be a natural question to ask: Is it consistent with ZF C that there exists a semiperfect right testing ring that is not right perfect? (2) As pointed out in the introduction if R = Z(p) is the localization of the ring of integers with respect to the prime ideal pZ, then R/J(R) is a testing ring while R is not. However, we wonder whether the R-testing property passes from R to R/J(R). Of course, by Theorem 2.7, this is the case if one assumes Shelah’s uniformization principle U Pk for sufficiently large k. Since there are rings that are right perfect but not left perfect, it is also natural to ask: Is the R-testing property left-right symmetric? Now the next few results are motivated by Example 2.9. Recall first that a ring R is called right P CI-ring if every proper cyclic right R-module is injective. Proposition 2.13. If R is a right P CI-ring, then every submodule of an Rprojective module M is R-projective. Proof. Let N be a submodule of an R-projective module M and consider the following diagram: i
→ M N f ↓ g π R → R/I → 0 where we may assume that I is a non-zero right ideal of R, i : N → M is the inclusion map and π : R → R/I is the canonical quotient map. Since R/I is injective as a right R-module, there is an R-homomorphism g : M → R/I such that g ◦ i = f . Since M is R-projective, we can find an R-homomorphism h : M → R such that π ◦ h = g. Now the map ϕ = h ◦ i : N → R satisfies π ◦ ϕ = g ◦ i = f , as required. 0
→
Proposition 2.14. If R is a right B-ring with a unique simple right R-module, then every right R-projective module M is a generator. Proof. Let T = T r(M ) = h(M ) denotes to the trace ideal of M in R. Clearly h:M →R
T is a 2-sided ideal of R, and we need to show that T = R. Assume to the contrary T ⊆ I where I is a maximal right ideal of R. Now, if N is a maximal submodule of σ M then M/N ∼ = R/I. Let f =: σ ◦ η where η : M → M/N is the canonical quotient map. Since M is R-projective, there is an R-homomorphism h : M → R such that π ◦ h = f where π : R → R/I is the quotient map. But this implies that Im h I, a contradiction. Proposition 2.15. If R is a right noetherian right V -domain of Krull dimension 1 with a unique simple right R-module, then every R-projective module M is torsionfree. Proof. Since R is a domain of right Krull dimension 1, it follows that R/I is of finite length, for every nonzero right ideal I of R. Inasmuch as R is a right V -ring, we infer that R/I is injective. Choose 0 = m ∈ M . By Proposition 2.13, the module
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N = mR is R-projective. Furthermore, by Proposition 2.14, R is a direct summand of N k for some k 1. If N were torsion, then N k , and hence R would also be torsion, a contradiction. Theorem 2.16. If R is a right principal ideal right V -domain with a unique simple right R-module, then every R-projective module M is locally free, and hence flat. Proof. Since right principal ideal domains have Krull dimension 1, we infer that R/I is of finite length, for every nonzero right ideal I of R. Since R is a right V -ring, R/I is injective, i.e. R is a right P CI-ring. Now by Proposition 2.13 and Lemma 2.1, it follows that every finitely generated submodule of M is projective. But every projective module over a principal ideal domain is free. Thus M is locally free, and hence flat, as required. Recall that the standard countably generated (flat) Bass’ module MR is the module with generators x1 , x2 , ... and relations x2 r2 = x1 , x3 r3 = x2 , ..., xi+1 ri+1 = xi , ...where 0 = ri ∈ R are non-units, i 1. Remark 2.17. (1) From what we have just proved in Theorem 2.16 above, if R is a right principal ideal right V -domain with a unique simple right R-module that is not right testing, then there exists a flat right R-module N which is R-projective but not projective. However, if R is the ring in Example 2.9, then the standard Bass’ module M over R is flat but not R-projective. To see this, note first that F ∼ = R/yR is the unique (up to isomorphism) simple right R-module. Since R is a domain, rR (x1 ) = 0. Now we can define the following non-zero R-homomorphism f : x1 R → F by sending x1 into 1 ∈ F ∼ = R/yR. By the injectivity of F ∼ = R/yR, f can be extended to a nonzero R-homomorphism g : M → F . We claim that g cannot be lifted to an R-homomorphism from M into R. This is because Hom(M, R) = 0. To see this, suppose that h : M → R is such a lifting of g. Since h(x1 ) ∈ Rrn ...r2 for every n 2, h(x1 ) = 0. Similarly, h(xn ) = 0 for every n 2, and so h = 0 as claimed. (2) Could one prove or disprove, in ZF C, the existence of a countably generated flat R-projective module which is not projective? 3. The Dual Baer Criterion While the above results and examples do not provide us with enough information to characterize right testing rings in ZF C, the next result may be considered as a positive step in dualizing the Baer Criterion. Indeed, since the injective hull of any module MR always exist and the dual statement for projective covers is equivalent to R being right perfect, the next theorem may be considered as the dual of the Baer Criterion. We start with the following well-known Lemma, where the equivalence, between (1) and (2) is due to H. Bass [2], and between (1) and (3) is due to F. L. Sandomierski [15]. Lemma 3.1. The following are equivalent: (1) R is (semi) perferct. (2) Every (finitely generated) right R-module has a projective cover. (3) Every (simple) semisimple right R-module has a projective cover.
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The next lemma is due to R. Ketkar and N. Vanaja [12]. Lemma 3.2. If R is a semiperfect ring, then every R-projective module with small radical is projective. In particular, over a perfect ring R, every R-projective right R-module is projective. Following the work of H. Bass [2] on projective covers, we call an R-epimorphism φ : P → M an R-projective cover of the R-module M if P is an R-projective module and ker φ P . Lemma 3.3. The following are equivalent: (1) R is semiperferct. (2) Every finitely generated right R-module has an R-projective cover. (3) Every simple right R-module has an R-projective cover. Proof. 1 ⇒ 2 ⇒ 3. Clear. 3 ⇒ 1. Let φ : P → S be an R-projective cover of the simple right R-module S. If K = ker φ, then K = rad(P ) P . Since P/rad(P ) ∼ = S, we may write P = L + rad(P ) for a cyclic submodule L of P . But since rad(P ) P , P = L is a cyclic module. Now, by Lemma 2.1, P is projective. This shows that every simple right R-module has a projective cover, and so R is semiperfect by Lemma 3.1. Theorem 3.4. The following are equivalent: (1) R is right perfect. (2) Every right R-module has a projective cover. (3) Every right R-module has an R-projective cover. (4) Every semisimple right R-module has an R-projective cover. Proof. 1 ⇒ 2 ⇒ 3 ⇒ 4. Clear. 4 ⇒ 1. By Lemma 3.3, R is semiperfect. If φ : P → S is an R-projective cover of the semisimple right R-module S, then by Lemma 3.2, P is projective, since rad(P ) P. Thus, every semisimple right R-module has a projective cover, and R is right perfect by Lemma 3.1. 4. Acknowledgments This research constitutes part of the Ph.D dissertation of the first author. Part of this work was carried out during a visit by the third author to the Ohio State University during the fall of 2014. He would like to express his gratitude to the members of the mathematics department at OSU for the financial support and the warm reception. The research of the fourth author was supported by the Mathematics Research Institute of the Ohio State University. References [1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, Berlin and New York, 1974. [2] H. Bass, Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. [3] R. Baer, Abelian groups that are direct summands of every containing abelian group, Bull. Amer. Math. Soc. 46 (1940), 800–806. [4] J.H. Cozzens, Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75-79. [5] H. Q. Dinh, C.J. Holton, D. V. Huynh, Quasi-projective modules over prime noetherian V -rings are projective or injective, J. Algebra 360 (2012), 87-91.
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¨ [6] B. Eckmann and A. Schopf, Uber injektive Moduln, Arch. Math. (Basel) 4 (1953), 75–78. [7] A. Facchini, Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Math., Vol. 187, Birkh¨ auser, 1998. [8] C. Faith, Algebra II. Ring Theory, Springer, 1976. [9] R.M. Hamsher, Commutative Noetherian Rings Over which every Module has a Maximal Submodule, Proc. Amer. Math. Soc. 17 (1966), 1471–1472. . [10] F. Kasch, Modules and Rings, L.M.S. Monograph No. 17, Academic Press, New York, 1982.F. [11] L.A. Koifman, Rings over which every module has a maximal submodule, Mat. Zametki 7 (1970), 359-367; (transl.) Math. Notes 7 (1970), 215-219. [12] R. D. Ketkar and N. Vanaja, R-projective modules over a semiperfect ring, Canad. Math. Bull. 24 (1981), 365-367. [13] L. Koifman, Rings over which each module has a maximal submodule, Mat. Zametki 3 (1970), 359-367. [14] S.H. Mohamed and B.J. M¨ uller, Continuous and Discrete Modules, Cambridge Univ. Press, Cambridge, UK, 1990. [15] F. L. Sandomierski, On semiperfect and perfect rings, Proc. Amer. Math. Soc. 21 (1969), 205-207. [16] F. Sandomierski, Relative injectivity and projectivity, PhD Thesis, Penn State Uniersity, 1964. [17] A. A. Tuganbaev, Rings over which each module has a maximal submodule. (Russian) Mat. Zametki 61 (1997), no. 3, 407–415; translation in Math. Notes 61 (1997), no. 3-4, 333–339. [18] J. Trlifaj, Whitehead test modules, Trans. Amer. Math. Soc. 348 (1996), 1521-1554. Department of Mathematics, Belarus State University, 220080, Minsk, Belarus. E-mail address: [email protected] Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. E-mail address: [email protected] Department of Mathematics, Belarus State University, 220080, Minsk, Belarus. E-mail address: [email protected] Department of Mathematics, The Ohio State University, Lima, Ohio 45804, USA. E-mail address: [email protected]
Hayder Abd Albaqir Alhilali, Yasser Ibrahim, Gena Puninski, Mohamed Yousif
PII: DOI: Reference:
S0021-8693(17)30250-8 http://dx.doi.org/10.1016/j.jalgebra.2017.04.010 YJABR 16197
To appear in:
Journal of Algebra
Received date:
7 January 2016
Please cite this article in press as: H.A.A. Alhilali et al., When R is a testing module for projectivity?, J. Algebra (2017), http://dx.doi.org/10.1016/j.jalgebra.2017.04.010
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
WHEN R IS A TESTING MODULE FOR PROJECTIVITY? HAYDER ABD ALBAQIR ALHILALI, YASSER IBRAHIM, GENA PUNINSKI, AND MOHAMED YOUSIF Abstract. A right R-module is called R-projective if it is projective relative to the right R-module RR . A ring R is called right testing for projectivity if every right R-projective module is projective. Every right perfect ring is right testing and there are examples of local rings that are not right testing. In this paper an attempt is made to understand the class of right testing rings and several examples are provided to highlight the set theoretic obstacles in characterizing such rings.
1. Introduction If M and N are right R-modules, then M is called N -projective (projective relative to N ) if every R-homomorphism from M into an image of N can be lifted to an R-homomorphism from M into N . M is called R-projective, if it is projective relative to the right R-module RR . The module M is called projective, if M is N -projective, for every right R-module N . Dually, we can define, N -injective, R-injective, and injective modules. Given any module M . An R-monomorphism σ : M → E is called an injective envelope (injective hull) of M if E is injective and σ(M ) ⊆ess E, i.e. K ∩ σ(M ) = 0, whenever 0 = K ⊆ E. Dually, an R-epimorphism φ : P → M is called a projective cover of the right R-module M , if P is projective and ker φ is small in P , i.e. L + ker φ = M for any proper submodule L of M . A result of Eckmann and Schopf [6] asserts that every right R-module M can be embedded in an injective envelope (hull) of M . In dualizing this result, Bass [2] has shown that every (finitely generated) right R-module has a projective cover if and only if R is a right (semi) perfect ring. On the other hand, a result of Baer [3] known by the Baer Criterion asserts that a right R-module M is injective if and only if it is injective relative to RR . In general, the dual to the Baer Criterion is not true as there are examples of R-projective modules that are not projective. While it is not difficult to see that every finitely generated R-projective module is projective (see Lemma 2.1 below), the situation is quite different if the module is not finitely generated. We will say that a ring R is right testing for projectivity if every right Rprojective module is projective. In his remarks at the end of chapter 22, [8], Faith left the characterization of right testing rings as an open problem. In [16], Sandomierski proved that every right perfect ring is a right testing ring. In [12], Ketkar and Vanaja extended the work of Sandomierski by showing that if R is a semiperfect ring then every R-projective module with small radical is projective. However, if R = Z(p) is the localization of the ring of integers with respect to the prime ideal 2010 Mathematics Subject Classification. Primary 16D40, 16D50, 16D60; Secondary 16L30, 16L60, 16P20, 16P40, 16P60. Key words and phrases. Projective modules, R-projective Modules, Baer Criterion, Bass rings. 1
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pZ, then R is a commutative noetherian valuation domain. In particular R is local and every nonzero ideal of R has the form pn R for some n. If M = R/pR is the unique simple R-module and E = E(M ) is the injective envelope of M , then clearly Hom(E, R/I) = 0 for every ideal I of R. Therefore E is R-projective but not projective. This shows that semiperfect (indeed local) rings need not be right testing. However, a result of Hamsher [9] asserts that if R is a commutative noetherian ring then R is a testing ring iff R is artinian, and in this paper we will show below that if R is a semilocal right noetherian ring, then R is right testing iff R is right artinian. Most recently, in [5, Theorem 3] the authors claimed that if R is a right hereditary right noetherian V -domain that is not artinian, then R is a right testing ring. Unfortunately the proof of this result contains a gap, and we will show below that such a result can not be carried out in the standard Zermelo-Frenkel axiomatic set theory with the axiom of choice, ZF C. Indeed, the work of Trlifaj in [18] shows that the characterization of right testing rings may depend on some set-theoretic axioms which are independent of ZF C. Throughout, rings R are associative with unity and modules are unitary right R-modules, unless specified otherwise. For a module M , we use rad(M ) and E(M ) to denote the Jacobson radical and the injective hull of M, respectively. If M = R, we write J(R) = rad(R). We write N ⊆ M if N is a submodule of M , N ⊆ess M if N is an essential submodule of M , and N ⊆⊕ M if N is a direct summand of M . We say that a submodule N of M is a small submodule and write N M if M = N + L for any proper submodule L M . We write M (I) for a direct sum of I-copies of M , M I for a direct product of I-copies of M, and Mod-R for the category of right R-modules. 2. On right hereditary right noetherian V -domains We start with the following observation. Lemma 2.1. Every finitely generated R-projective module M is projective. η
Proof. Let R(n) −→ M → 0 be the natural R-epimorphism with n 1. By [1, Theorem 16.12], since M is R-projective, M is R(n) -projective and the map η splits. Therefore M is projective. Definition 2.2. A ring R is called right testing for projectivity (or just right testing) if every right R-projective module is projective. As we have seen in the proof of Lemma 2.1 above, a right R-module M is Rprojective if and only if M is R(n) -projective, for every n 1. Now the next result is a straightforward application of [1, Proposition 21.6]. Proposition 2.3. If R and S are Morita equivalent rings, then R is right testing if and only if S is right testing. While it will be interesting to look for other consistent set theoretic universes where right testing but not right perfect rings exist, we will limit our work below within the well-established ZF C with some additional mild set-theoretic axioms that are consistent with ZF C. In this direction, we will put to use the following theory developed by Trlifaj in [18]: A class of (right) R-modules M is said to be ptesting if whenever N is a right R-module such that Ext(N, M ) = 0 for every right R-module M ∈ M then N is projective. For instance, the class of all R-modules is
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p-testing. If k is a cardinal of cofinality ℵ0 , Trlifaj showed in [18, Lemma 2.4] how to use Shelah’s uniformization principle U Pk (which is consistent with ZF C + GCH) to disprove the existence of proper p-testing classes by establishing the next result. For a definition of Shelah’s uniformization principle U Pk , where k is a cardinal of cardinality ℵ0 , see [18, Page 1526]. Lemma 2.4. Suppose that R is not a right perfect ring and let k be an uncountable cardinal of cofinality ℵ0 . If U Pk holds, then there exists a non-projective right Rmodule N such that Ext(N, M ) = 0 for every right R-module M of cardinality < k. The non-projective module N in the proof of Lemma 2.4 above is a flat module of projective dimension 1. Furthermore if k |R| + ℵ0 , then N is k-generated. In the next proposition we provide a sufficient condition for an R-module to be R-projective. Proposition 2.5. If M is a right R-module such that Ext(M, I) = 0 for every non-zero proper right ideal I of R, then M is R-projective. In particular, if R is a right testing ring then the class of proper right ideals of R is p-testing. Proof. By applying Hom(M, −) to the short exact sequence 0 → I → R → R/I → 0, we obtain the following exact sequence: 0 → Hom(M, I) → Hom(M, R) → Hom(M, R/I) → Ext(M, I) → Ext(M, R) → ... If Ext(M, I) = 0 for every proper right ideal I of R, it follows that M is Rprojective. The last assertion is clear if R is right testing. In the next result we establish a converse to Proposition 2.5 in the case of selfinjective rings. Proposition 2.6. Let R be a right self-injective ring and M a right R-module. Then M is R-projective if and only if Ext(M, I) = 0 for every right ideal I of R. Proof. Consider the long exact sequence in the proof of Proposition 2.5 above. Since R is right self-injective, Ext(M, R) = 0. Then the map Hom(M, R) → Hom(M, R/I) is onto if and only if Ext(M, I) = 0 for every right ideal I of R. In the next theorem we show that right testing rings are right perfect if one assumes Shelah’s uniformization principle U Pk . Theorem 2.7. Let R be a ring of cardinality λ and let k = (λ+ )ω . If U Pk is assumed, then R is right testing if and only if R is right perfect. Proof. It sufficies to check only the forward implication. So assume that R is right testing. If R is not right perfect, then by Lemma 2.4, there exists a non-projective right R-module N such that Ext(N, I) = 0 for every proper right ideal I of R. Now, by Proposition 2.5, we infer that N is projective, a contradiction. Remark 2.8. In [5, Theorem 3] the authors claimed that if R is a right hereditary right noetherian V -domain that is not artinian, then R is a right testing ring, where a ring R is called a right V -ring (after Villamayor) if every simple right R-module is injective. Unfortunately there is a gap in the second pragraph of the proof of this result. However, since such a ring is not right perfect, assuming (sufficiently large) U Pk , we infer from Theorem 2.7 that the ring R cannot be right testing. Thus the
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proof of [5, Theorem 3] cannot be carried out in ZF C. Of course this does not prevent [5, Theorem 3] to be true in other ZF C extensions. Indeed, we have no answer to the following general question: Is it consistent with ZF C that there exists a right testing ring that is not right perfect? The following is a specific example of a hereditary noetherian V -domain, due to Cozzens [4] (see also [11]) that is neither left nor right testing ring if one assumes Shelah’s uniformization principle U Pk . Example 2.9. Let K be a universal differential field with a derivation δ and let R = K [y; δ] denote the ring of differential polynomials in the indeterminate y with coefficients in K, where addition is defined in the obvious way and multiplication defined by yk = ky + δ(k) for all k ∈ K. Then R is a right (and left) principal ideal domain. R is also a right (and left) P CI-ring; i.e. R/I is injective whenever I is a nonzero right (left) ideal of R. In particular, simple right (and left) R-modules are injective (i.e. R is right and left V -ring). Observe that the trivial module K is the unique (up to isomorphism) simple module of R and R/I ∼ = K n , for some n, whenever I is a nonzero right (left) ideal of R. If one assumes K to be countable, then the ring R = K [y; δ] is also countable and is neither right nor left perfect ring. If Shelah’s uniformization principle U Pk for k = (ℵ1 )ω is assumed, then R is neither left nor right testing. Recall that a ring R is called a right B-ring (the B stands for Bass) if every nonzero right R-module contains a maximal submodule. It is known that, R is a right B-ring if and only if J(R) is left T -nilpotent and R/J(R) is a right B-ring, see for example [17, Lemma 1]. Furthermore, Koifman, [13, Theorem 3], characterized commutative B-rings as those rings whose J(R) is T -nilpotent and R/J(R) is a von Neumann regular ring. For Noetherian rings we have the following. Lemma 2.10. Every right noetherian right testing ring R is a right B-ring. In particular J(R) is nilpotent. Proof. Assume to the contrary that there exists a nonzero right R-module M with no maximal submodules. We claim that Hom(M, R/I) = 0 for every right ideal I of R. Otherwise, let f : M → R/I be a nonzero R-homomorphism. Since f (M ) is a submodule of the right noetherian R-module R/I, it contains a maximal submodule N . But then f −1 (N ) is a maximal submodule of M , a contradiction. Thus M is R-projective, and hence projective since R is right testing. Since every projective module contains a maximal submodule, we obtain another contradiction. The last assertion follows from the above remarks and Levitzky’s theorem which states that if a ring R is right neoetherian then every one-sided nil ideal is nilpotent (see [1, Theorem 15.22]). A result of Hamsher [9] asserts that if R is a commutative noetherian ring then R is a testing ring if and only if R is artinian. In the next proposition we obtain a noncommutative version of Hamsher’s result for semilocal right noetherian rings. Proposition 2.11. Let R be a semilocal right noetherian ring. Then R is right testing if and only if R is right artinian. Proof. We only need to prove the forward implication (⇒). By Lemma 2.10, since R is right noetherian, J(R) is nilpotent. Thus R is a semiprimary ring, and hence right artinian.
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Since every right noetherian right testing ring is a right B-ring and every commutative B-ring R has R/J(R) regular, Hamsher’s result is now an immediate consequence of Proposition 2.11. Remark 2.12. (1) The following seems to be a natural question to ask: Is it consistent with ZF C that there exists a semiperfect right testing ring that is not right perfect? (2) As pointed out in the introduction if R = Z(p) is the localization of the ring of integers with respect to the prime ideal pZ, then R/J(R) is a testing ring while R is not. However, we wonder whether the R-testing property passes from R to R/J(R). Of course, by Theorem 2.7, this is the case if one assumes Shelah’s uniformization principle U Pk for sufficiently large k. Since there are rings that are right perfect but not left perfect, it is also natural to ask: Is the R-testing property left-right symmetric? Now the next few results are motivated by Example 2.9. Recall first that a ring R is called right P CI-ring if every proper cyclic right R-module is injective. Proposition 2.13. If R is a right P CI-ring, then every submodule of an Rprojective module M is R-projective. Proof. Let N be a submodule of an R-projective module M and consider the following diagram: i
→ M N f ↓ g π R → R/I → 0 where we may assume that I is a non-zero right ideal of R, i : N → M is the inclusion map and π : R → R/I is the canonical quotient map. Since R/I is injective as a right R-module, there is an R-homomorphism g : M → R/I such that g ◦ i = f . Since M is R-projective, we can find an R-homomorphism h : M → R such that π ◦ h = g. Now the map ϕ = h ◦ i : N → R satisfies π ◦ ϕ = g ◦ i = f , as required. 0
→
Proposition 2.14. If R is a right B-ring with a unique simple right R-module, then every right R-projective module M is a generator. Proof. Let T = T r(M ) = h(M ) denotes to the trace ideal of M in R. Clearly h:M →R
T is a 2-sided ideal of R, and we need to show that T = R. Assume to the contrary T ⊆ I where I is a maximal right ideal of R. Now, if N is a maximal submodule of σ M then M/N ∼ = R/I. Let f =: σ ◦ η where η : M → M/N is the canonical quotient map. Since M is R-projective, there is an R-homomorphism h : M → R such that π ◦ h = f where π : R → R/I is the quotient map. But this implies that Im h I, a contradiction. Proposition 2.15. If R is a right noetherian right V -domain of Krull dimension 1 with a unique simple right R-module, then every R-projective module M is torsionfree. Proof. Since R is a domain of right Krull dimension 1, it follows that R/I is of finite length, for every nonzero right ideal I of R. Inasmuch as R is a right V -ring, we infer that R/I is injective. Choose 0 = m ∈ M . By Proposition 2.13, the module
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N = mR is R-projective. Furthermore, by Proposition 2.14, R is a direct summand of N k for some k 1. If N were torsion, then N k , and hence R would also be torsion, a contradiction. Theorem 2.16. If R is a right principal ideal right V -domain with a unique simple right R-module, then every R-projective module M is locally free, and hence flat. Proof. Since right principal ideal domains have Krull dimension 1, we infer that R/I is of finite length, for every nonzero right ideal I of R. Since R is a right V -ring, R/I is injective, i.e. R is a right P CI-ring. Now by Proposition 2.13 and Lemma 2.1, it follows that every finitely generated submodule of M is projective. But every projective module over a principal ideal domain is free. Thus M is locally free, and hence flat, as required. Recall that the standard countably generated (flat) Bass’ module MR is the module with generators x1 , x2 , ... and relations x2 r2 = x1 , x3 r3 = x2 , ..., xi+1 ri+1 = xi , ...where 0 = ri ∈ R are non-units, i 1. Remark 2.17. (1) From what we have just proved in Theorem 2.16 above, if R is a right principal ideal right V -domain with a unique simple right R-module that is not right testing, then there exists a flat right R-module N which is R-projective but not projective. However, if R is the ring in Example 2.9, then the standard Bass’ module M over R is flat but not R-projective. To see this, note first that F ∼ = R/yR is the unique (up to isomorphism) simple right R-module. Since R is a domain, rR (x1 ) = 0. Now we can define the following non-zero R-homomorphism f : x1 R → F by sending x1 into 1 ∈ F ∼ = R/yR. By the injectivity of F ∼ = R/yR, f can be extended to a nonzero R-homomorphism g : M → F . We claim that g cannot be lifted to an R-homomorphism from M into R. This is because Hom(M, R) = 0. To see this, suppose that h : M → R is such a lifting of g. Since h(x1 ) ∈ Rrn ...r2 for every n 2, h(x1 ) = 0. Similarly, h(xn ) = 0 for every n 2, and so h = 0 as claimed. (2) Could one prove or disprove, in ZF C, the existence of a countably generated flat R-projective module which is not projective? 3. The Dual Baer Criterion While the above results and examples do not provide us with enough information to characterize right testing rings in ZF C, the next result may be considered as a positive step in dualizing the Baer Criterion. Indeed, since the injective hull of any module MR always exist and the dual statement for projective covers is equivalent to R being right perfect, the next theorem may be considered as the dual of the Baer Criterion. We start with the following well-known Lemma, where the equivalence, between (1) and (2) is due to H. Bass [2], and between (1) and (3) is due to F. L. Sandomierski [15]. Lemma 3.1. The following are equivalent: (1) R is (semi) perferct. (2) Every (finitely generated) right R-module has a projective cover. (3) Every (simple) semisimple right R-module has a projective cover.
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The next lemma is due to R. Ketkar and N. Vanaja [12]. Lemma 3.2. If R is a semiperfect ring, then every R-projective module with small radical is projective. In particular, over a perfect ring R, every R-projective right R-module is projective. Following the work of H. Bass [2] on projective covers, we call an R-epimorphism φ : P → M an R-projective cover of the R-module M if P is an R-projective module and ker φ P . Lemma 3.3. The following are equivalent: (1) R is semiperferct. (2) Every finitely generated right R-module has an R-projective cover. (3) Every simple right R-module has an R-projective cover. Proof. 1 ⇒ 2 ⇒ 3. Clear. 3 ⇒ 1. Let φ : P → S be an R-projective cover of the simple right R-module S. If K = ker φ, then K = rad(P ) P . Since P/rad(P ) ∼ = S, we may write P = L + rad(P ) for a cyclic submodule L of P . But since rad(P ) P , P = L is a cyclic module. Now, by Lemma 2.1, P is projective. This shows that every simple right R-module has a projective cover, and so R is semiperfect by Lemma 3.1. Theorem 3.4. The following are equivalent: (1) R is right perfect. (2) Every right R-module has a projective cover. (3) Every right R-module has an R-projective cover. (4) Every semisimple right R-module has an R-projective cover. Proof. 1 ⇒ 2 ⇒ 3 ⇒ 4. Clear. 4 ⇒ 1. By Lemma 3.3, R is semiperfect. If φ : P → S is an R-projective cover of the semisimple right R-module S, then by Lemma 3.2, P is projective, since rad(P ) P. Thus, every semisimple right R-module has a projective cover, and R is right perfect by Lemma 3.1. 4. Acknowledgments This research constitutes part of the Ph.D dissertation of the first author. Part of this work was carried out during a visit by the third author to the Ohio State University during the fall of 2014. He would like to express his gratitude to the members of the mathematics department at OSU for the financial support and the warm reception. The research of the fourth author was supported by the Mathematics Research Institute of the Ohio State University. References [1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, Berlin and New York, 1974. [2] H. Bass, Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. [3] R. Baer, Abelian groups that are direct summands of every containing abelian group, Bull. Amer. Math. Soc. 46 (1940), 800–806. [4] J.H. Cozzens, Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75-79. [5] H. Q. Dinh, C.J. Holton, D. V. Huynh, Quasi-projective modules over prime noetherian V -rings are projective or injective, J. Algebra 360 (2012), 87-91.
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¨ [6] B. Eckmann and A. Schopf, Uber injektive Moduln, Arch. Math. (Basel) 4 (1953), 75–78. [7] A. Facchini, Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Math., Vol. 187, Birkh¨ auser, 1998. [8] C. Faith, Algebra II. Ring Theory, Springer, 1976. [9] R.M. Hamsher, Commutative Noetherian Rings Over which every Module has a Maximal Submodule, Proc. Amer. Math. Soc. 17 (1966), 1471–1472. . [10] F. Kasch, Modules and Rings, L.M.S. Monograph No. 17, Academic Press, New York, 1982.F. [11] L.A. Koifman, Rings over which every module has a maximal submodule, Mat. Zametki 7 (1970), 359-367; (transl.) Math. Notes 7 (1970), 215-219. [12] R. D. Ketkar and N. Vanaja, R-projective modules over a semiperfect ring, Canad. Math. Bull. 24 (1981), 365-367. [13] L. Koifman, Rings over which each module has a maximal submodule, Mat. Zametki 3 (1970), 359-367. [14] S.H. Mohamed and B.J. M¨ uller, Continuous and Discrete Modules, Cambridge Univ. Press, Cambridge, UK, 1990. [15] F. L. Sandomierski, On semiperfect and perfect rings, Proc. Amer. Math. Soc. 21 (1969), 205-207. [16] F. Sandomierski, Relative injectivity and projectivity, PhD Thesis, Penn State Uniersity, 1964. [17] A. A. Tuganbaev, Rings over which each module has a maximal submodule. (Russian) Mat. Zametki 61 (1997), no. 3, 407–415; translation in Math. Notes 61 (1997), no. 3-4, 333–339. [18] J. Trlifaj, Whitehead test modules, Trans. Amer. Math. Soc. 348 (1996), 1521-1554. Department of Mathematics, Belarus State University, 220080, Minsk, Belarus. E-mail address: [email protected] Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. E-mail address: [email protected] Department of Mathematics, Belarus State University, 220080, Minsk, Belarus. E-mail address: [email protected] Department of Mathematics, The Ohio State University, Lima, Ohio 45804, USA. E-mail address: [email protected]