ON ALMOST PRECOVERS AND ALMOST PREENVELOPES*

Available online at www.sciencedirect.com sciE*cE@DiREcT*

Acta Mathematica Scientia 2006,26B(3):395-400

B%B@+@ www.wipm.ac.cn/publish/

ON ALMOST PRECOVERS AND ALMOST PREENVELOPES* Ma0 Lixin ( %*% ) Department of Basic Courses, Nanjing Institute of Technology, Nanjing 211167, China

Ding Nanqing ( T#)& )

Tong Wenting ( %kE)

Department of Mathematics, Nanjing University, Nanjing 210093, China

Abstract Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost Cprecover and the direct sum of an almost C-cover and a weak C-cover is a weak C-cover. Then the notion of almost C-preenvelopes is introduced and studied. Key words Almost precover, almost preenvelope 2000 MR Subject Classification

1

16D10, 16D90

Introduction

Throughout this article, R is an associative ring with identity and all modules are unitary right R-modules. N 5 M , N 5, M and N << M means that N is a submodule, an essential submodule, and a superfluous submodule of M , respectively. The class C of R-modules are assumed to contain 0 and to be closed under isomorphisms and finite direct sums. General background materials can be found in [l],[2]. Recall that an R-homomorphism qh : M --f F with F E C is called a C-preenvelope of a module M (see [3]), if for any R-homomorphism f : M -+ F' where F' E C there is an R-homomorphism g : F -+ F' such that g 4 = f. If, furthermore, when F' = F and f = 4, the only such g is a n automorphism of F , then 4 is called a C-envelope of M . If C is the class of injective modules, then we get the usual injective envelopes. If envelopes exist, they are unique up to isomorphism. Dually we have the concepts of C-precovers and C-covers. Recently, the notions of almost C-(pre)covers and weak C-covers were introduced in [4] as generalizations of (pre)covers. In the present article, we first show that the finite direct sum of weak C-precovers is a weak C-precover and the direct sum of a n almost C-cover and a weak C-cover is a weak C-cover. Then the notion of almost C-preenvelopes is defined as the dual of almost C-precovers and some properties are given. 'Received May 26, 2003. This research was partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (20020284009, 20030284033), NSF of China (10331030), Jiangsu Planned Projects for Postdoctoral Research Funds (0203003403), the Postdoctoral Research Funds of China (2005037713), and the Nanjing Institute of Technology of China.

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Properties of Almost Precovers

Recall that an R-module homomorphism f : M -+ N is an essential monomorphism if f is monic and imf 5, N ; f is a superfluous epimorphism if f is epic and kerf << M (see [l]). Following [4], an R-homomorphism 'p : X -+ M with X E C is called an almost C-precover of M , if for each F E C and each R-homomorphism f : F +. M there is an essential submodule F' of F with F' E C and an R-homomorphism g : F' -+ X such that 'pg = f L , where L : F' -+ F is the inclusion map. It is easy to see that 'p : X + A4 is an almost C-precover if and only if for each F E C and each R-homomorphism f : F -+ M , there is an essential monomorphism ?(, : E 4 F with E E C and an R-homomorphism g : E X such that 'pg = fq. Lemma 2.1 Consider the following pullback diagram: B ---f

A - B

where A = {(c, b)lf(c) = g ( b ) , c E C,b E B } , a ( c ,b) = c, P(c, b ) = b. (1) If g is an essential monomorphism and f is a monomorphism, then a is an essential monomorphism. (2) Iff and g are both essential monomorphisms, then a and p are both essential monomorphisms. Proof (1) Because g is a monomorphism, a is also a monomorphism by the property of a pullback. So it is enough to show that a is essential. Suppose there exists 0 # N 5 C such that (ima) nN = 0. As f is a monomorphism, f ( N ) # 0. It follows that (img) n f ( N ) # 0 as g is an essential monomorphism. Hence there exist 0 # a E N , 0 # b E B such that f(a) = g ( b ) # 0, and so ( a ,b) E A. As a ( a ,b ) = a , a E (ima) n N = 0. So a = 0 is a contradiction. (2) follows from (1). Recall that a class C of modules is called weakly hereditary [4]if for any 0 # M E C, every nonzero submodule of M contains an essential submodule from C. It is well known that if 'pi : X i -+ Mi,i = 1 , 2 , . . . , n are C-precovers, then e'pi: @Xi $Mi is a C-precover [2]. Here we have the following Theorem 2.2 Let C be weakly hereditary. If pi : Xi -+ Mi, i = 1 , 2 , . . . ,n are almost C-precovers, then @(pi : exi e M i is an almost C-precover. Proof It is enough to show the case n = 2. Suppose f : D -+ MI @ Mz is any Rhomomorphism with D E C. Let ~i : M I @ M2 Mi be the canonical projection, i = 1 , 2 . As each (pi : X i -+ Mi is an almost C-precover, there exist Bi 5, D with Bi E C and Rhomomorphisms 7Ji : Bi + X i such that the following diagram is commutative: -+

.---f

---f

Bz

D

where ~i : Bi -+ D is the inclusion mapping, i = 1 , 2 . Consider the following pullback diagram:

No.2

Mao

e~ ai:

ON ALMOST PRECOVERS AND ALMOST PREENVELOPES 0

By Lemma 2.1, Q: is essential. So L ~ Q := L Z / ~: A

397

0

---f

D is essential by [l,Exercise 5.141. Let a E A.

ww)

= ( 9 1 @ (Pz)($1Q:(a),$zP(a))= (cpl!blQ:(a), cp2+2P(a)) = zf~zP(u =)~)L I Q : ( ( L ) . Thus (PI@cpz)($ia@ $ 2 p ) = f ~ 1 aAs . C is weakly hereditary, there exists A' 5, A with A' E C. So we have the following commutative diagram:

Note that

(91 @ ( P Z ) ( W @

(TI f L I Q : ( B ) ,~

1,Ql

A'

(w,O@W,

,D

511

If

where X : A' -+ A is the inclusion. Note that ~ 1 a Xis an essential monomorphism, therefore cpi @ cp2 : XI@ X2 -+ M I @ Mz is an almost C-precover, as desired. Recall that an almost C-precover cp : G -+ M of a module M is called a weak C-cover [4] if each endomorphism f of G with cpf = cp is an essential monomorphism, and cp is called an almost C-cover [4]if each endomorphism f of G with cpf = cp is an automorphism of G.

It is known that if (pi : Xi is a C-cover [4]. Here we have

-+

Mi,i = I,2 are C-covers, then c p l @ cp2 : X i @ X2

-+

M I @ Mz

Theorem 2.3 Let C be weakly hereditary. If (PI : X1 --+ M I is an almost C-cover, MZ is a weak C-cover, then (PI @ cpz : X I @ X Z -+ M I @ MZ is a weak C-cover.

(pz : Xz

-+

Proof By Theorem 2 . 2 , cp1 @ cp2 : X I @ X ,

-+ M I @ A42

is an almost C-precover. Now

suppose that f is an endomorphism of X1 @ X Z such that cp1 @ cpz = ( 9 1 @ cpz)f. We shall show that f is an essential monomorphism. Let 7ri : X I @ Xz Xi be the canonical projection ---f

and

Li

:

Xi

-+

X1 @ X2 the canonical injection, i = 1,2. For convenience we express the

elements in X1 @ X Z as columns

cp1

for

21

E X I ,22 E

fll

(PI @ cpz =

cpz)f means that wfll = (p1,cplf12 = 0, 'pzfz1 = 0, is an automorphism of X I . Consider the matrix equation

@ cp2 = (cpl @

hypothesis,

Xz. Then

+

(pzf22

=

~

2

+

By .

As cpzf21 = 0, cpzf2z = cpz, we get c p z ( - f z 1 f i i 1 f 1 2 f z 2 ) = cpz. Hence -fzlfiilflz f2z is an essential monomorphism by hypothesis. Now by a standard matrix argument we see that f is monic. So the proof is complete if we show that imf 5 , X I @ X Z . Let g =

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f12

0

-fz1fii1f12

. We claim that img 5, XI CB XZ. In fact, im(f1l

+ f22

Vo1.26 Ser.B

&, (-

fzl fG'fl2

+

\

+fzz)) 5, Xi €8 X z by [I,Proposition 5.201, and so img S e X 1 @ Xz. Thus imf 5, X1@ X z .

3

Almost Preenvelopes

In this section, the concept of almost preenvelopes is introduced as the dual of almost precovers. We start with the following Definition 3.1 Let C be a class of modules. A homomorphism : M 4 G with G E C is called an almost C-preenvelope of M , if for each F E C and each homomorphism f : M -+ F there are superfluous epimorphism 7r : F -+ F' with F' E C and a homomorphism g : G F' such that gg5 = 7rf . An almost C-preenvelope cp : M + G with G E C is called a weak C-envelope if each endomorphism f of G with f c p = cp is a superfluous epimorphism, and cp is called an almost C-envelope if each endomorphism f of G with f cp = cp is an automorphism of G. C is said to be weakly homomorphically closed, if for any A E C and any epimorphism A 4 B there exists C << B such that B/C E C. Theorem 3.2 Let C be a weakly homomorphically closed class of modules and the following diagram I$

-+

M+

G

be a pushout diagram. If cp is an almost C-preenvelope and x : G --f N a superfluous epimorphism with N E C, then x11,is an almost C-preenvelope. Proof Let H E C and a : M -+ H be an R-homomorphism. As cp is an almost C-preenvelope, there exists a superfluous epimorphism /3 : H -+ L with L E C and an Rhomomorphism y : F' -+ L such that ycp = P ( a f ) = ( P a ) f . So by the property of a pushout, there exists g5 : G -+ L such that the following diagram is commutative:

Thus g5+ = pa. Let K = L/g5(kerr), p : L K be the canonical map. As 7r is a superfluous epimorphism, $(ker7r) << L. Hence p is a superfluous epimorphism. Let S : N -+ K be the induced -+

No.2

Mao et al: ON ALMOST PRECOVERS AND ALMOST PREENVELOPES

399

homomorphism. Then we have the following commutative diagram:

.i7 P

f6

G+K

As C is weakly homomorphically closed, there exists a superfluous epimorphism 8 : K with E E C. Thus we get the following commutative diagram:

---f

E

Note that OpP is a superfluous epimorphism by [l,Exercises 5.141. So m) is an almost Cpreenvelope.

Corollary 3.3 Let C be a weakly homomorphically closed class of modules. Then every module has an almost C-preenvelope if and only if every flat module has an almost C-preenvelope. Proof One direction is obvious. Now suppose every flat module has an almost Cpreenvelope. Let M be any R-module. By [ 5 ] , M has a flat cover a : F ( M ) -+ M . It L by hypothesis. Consider the follows that F ( M ) has an almost C-preenvelope 4 : F ( M ) following pushout diagram: ---f

Note that a is epic, so is p. As C is weakly homomorphically closed, there is a superfluous epimorphism 7r : N -+ H with H E C. Thus T$J is an almost C-preenvelope by Theorem 3.2. Lemma 3.4 Consider the following pushout diagram:

B

-

-

where M = (C f q / { ( a ( a ) ,-P(a))l a E A ) , f(c) = (c, O ) , g ( b ) = (0, b). 1. If a is a superfluous epimorphism and /3 an epimorphism, then g is a superfluous epimorphism. 2. If a and P are both superfluous epimorphisms, then f and g are both superfluous epimorphisms. Proof (1)As a is an epimorphism, g is also an epimorphism by the property of a pushout. So it is enough to show that g is superfluous. Let kerg+N = B with N 5 B. We first claim that kera+,B-l(N) = A. In fact, let a E A. Then there exist z E kerg, y E N such that p(a) = z+y. As P is epic, there exists s E A such that p ( s ) = y. Note that gp(a) = g(z) +g(y) = g p ( s ) , and so gp(u - s) = 0. Thus f a ( a - s ) = gP(a - s) = 0, that is, ( a ( a - s),0) = 0. Hence there exists t E A such that a ( u - s ) = a(t) and 0 = -p(t). Note that a - s - t E kera, s t E P-l(N)

+

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and so a = ( a - s - t ) + (s + t ) E kercr+D-l(N), therefore kera+p-'(N) = A . It follows that P - ' ( N ) = A because a is superfluous. Thus N = B,as required. (2) follows from ( I ) . We omit the proofs of the following two results that are dual to those of Theorems 2.2 and Theorem 2.3 using Lemma 3.4 in place of Lemma 2.1. Theorem 3.5 Let C be weakly homomorphically closed. If (pi : Mi Gi, i = 1 , 2 , . . . ,n are almost C-preenvelopes, then $(pi : @Mi @Gi is an almost C-preenvelope. Theorem 3.6 Let C be weakly homomorphically closed. If cp1 : M I + G1 is an almost C-envelope, (p2 : MZ -+ G2 is a weak C-envelope, then cp1$ (PZ : MI @ M2 + GI @ G2 is a weak C-envelope. --f

--f

References 1 Anderson F W, Fuller K R. Rings and Categories of Modules. New York: Springer-Verlag. 1974 2 Xu J. Flat Covers of Modules. In: Lecture Notes in Mathematics 1634. Berlin, Heidelberg, New York Springer Verlag, 1996 3 Enochs E E. Injective and flat covers, envelopes and resolvents. Israel J Math, 1981, 39: 189-209 4 Bican L, Torrecillas B. Almost precovers. Comm Algebra, 2002, 30: 477-487 5 Bican L, Bashir E, Enochs E E. All modules have flat covers. Bull London Math SOC,2001, 33: 385-390