LU BO

(College of Mathematics and Information Science,Northwest Normal University, Lanzhou,730070)

F-perfect Rings and Modules*

LU BO

(College of Mathematics and Information Science,Northwest Normal University, Lanzhou,730070)

Communicated by Du Xian-kun

Let R be a ring,and let(F,C)be a cotorsion theory.In this article,the notion of F-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings.A ring R is said to be right F-perfect if F is projective relative to R for any F∈F.We give some characterizations of F-perfect rings.For example, we show that a ring R is right F-perfect if and only if F-covers of f i nitely generated modules are projective.Moreover,we def i ne F-perfect modules and investigate some properties of them.

F-perfect ring,F-cover,F-perfect module,cotorsion theory,projective module

1 Introduction

In 1953,Eckmann and Schopf[1]proved the existence of injective envelopes of modules over any associative ring.The dual problem,that is,the existence of projective covers was studied by Bass[2]in 1960.In spite of the existence of injective envelopes over any ring, he proved that over a ring R,all right modules have projective covers if and only if R is a right perfect ring.In[3],a ring R is called right almost-perfect if every f l at right R-module is projective relative to R,and proved that a ring is right almost-perfect if and only if f l at covers of f i nitely generated modules are projective.In this article,we introduce the concept of F-perfect rings.We give some characterizations of F-perfect rings.For example,we show that a ring R is right F-perfect if and only if F-covers of f i nitely generated modules are projective.

Let X be a class of R-modules.We denote

A pair(F,C)of classes of R-modules is called a cotorsion theory if F⊥=C and F=⊥C (see[4]).A cotorsion theory(F,C)is called complete if every R-module has a special C-preenvelope(and a special F-precover)(see[5]).A cotorsion theory(F,C)is called perfect if every R-module has a C-envelope and an F-cover(see[6,7]).A cotorsion theory(F,C) is said to be hereditary if whenever 0→L′→L→L′′→0 is exact with L,L′′∈F then L′is also in F,or equivalently,if 0→C′→C→C′′→0 is exact with C′,C∈C then C′′is also in C(see[8]).

Let R be a ring and C be a class of R-modules which is closed under isomorphic copies. A C-precover of an R-module M is a homomorphism φ:F→M with F∈C such that for any homomorphism ψ:G→M with G∈C,there existsμ:G→F such that φμ=ψ.A C-precover φ:F→M is said to be a C-cover if every endomorphism λ of F with φλ=φ is an automorphism of F.Dually,a C-preenvelope and a C-envelope of an R-module are def i ned.

In[4]a ring R is called right almost-perfect if every f l at right R-module is projective relative to R;equivalently,f l at covers of f i nitely generated right R-modules are projective. It was shown that right perfect rings are right almost-perfect,and right almost-perfect rings are semiperfect,but not conversely.In Section 2,we introduce the notion of F-perfect rings as a generalization of the notion of almost-perfect rings,that is,we call a ring R F-perfect in case F is projective relative to R for any F ∈F.We give some characterizations of F-perfect rings.For example,in Theorem 2.1 we show that a ring R is right F-perfect if and only if F-covers of f i nitely generated modules are projective.And in Theorem 2.3 we prove that a ring R is right F-perfect if and only if for every right R-modules F with F∈F, if

where P is a f i nitely generated projective summand of F and U≤F,thenfor some

In Section 3,we introduce the notion of F-perfect modules,that is,let(F,C)be a perfect cotorsion theory.We call an R-module M F-perfect in case the F-cover of every factor module of M is projective.We show that F-perfectness is closed under factor modules, extensions,and f i nite direct sums.Also some characterizations of F-perfect modules are given.

Throughout this article,all rings are associative with identity,and all modules are unitary right modules unless stated otherwise.For a ring R,let J=J(R)be the Jacobson radical of R.(F,C)denotes a cotorsion theory.F(resp.,C)denotes the F(resp.,C)of the cotorsion theory(F,C)unless stated otherwise.

General background materials can be found in[4,9–10].

2 F-perfect Rings

Let R be a ring,and(F,C)be a cotorsion theory.

Lemma 2.1[11]Let U be an R-module.

(1)If 0→ M′→ M → M′′→ 0 is an exact sequence of R-modules and U is M-projective,then U is projective relative to both M′and M′′.

(2) If U is projective relative to each R-module Mi(1≤ i≤ n),then U isprojective.

Moreover,if U is f i nitely generated and Mα(α∈A),then U is projective relative to⊕AMα.

Def i nition 2.1 Let(F,C)be a cotorsion theory.A ring R is called right F-perfect if every right R-module F with F ∈F is projective relative to R.Left F-perfect rings are def i ned similarly.If R is both left and right F-perfect,then R is called an F-perfect ring.

Remark 2.1 Let R be a ring.

(1)Let F be the class of f l at right R-modules.Then R is F-perfect if and only if R is A-perfect.

(2)Let F be the class of right R-modules of f l at dimension at most n.Then F-perfect rings are A-perfect,but A-perfect rings are not necessarily F-perfect.

(3)Let F be the class of n-f l at right R-modules.If R is A-perfect,then R is F-perfect (since(Fn,Cn)is a complete hereditary cotorsion,where Fn(resp.,Cn)denotes the class of modules all n-f l at(resp.,n-cotorsion)right R-modules.And n-f l at right R-modules is f l at (see[12])).But if R is F-perfect,then R is not necessarily A-perfect.

(4)Let R be a right coherent ring,and

where FPnis the class of all right R-modules of FP-injective dimension at most n.Then (FPn,F)is a perfect cotorsion theory(see[13]).But A-perfect rings are not necessarily F-perfect and F-perfect rings are not necessarily A-perfect.

Lemma 2.2 Let(F,C)be a cotorsion theory,and φ:F→ M be an F-cover of the R-module M.If F is projective,then φ:F→M is a projective cover of M.

Proof.Since φ:F→M is an F-cover of the R-module M,φ is an epimorphism.Now let L+kerφ=F with L≤F.So φ|L:L→M is an epimorphism.By the projectivity of F, there is λ:F→L?F such that

Since φ:F→M is an F-cover of the R-module M,λ is an automorphism of F,and hence

Therefore,

and so φ:F→M is a projective cover of M.

Lemma 2.3[10]Let f:F→M be an F-cover of the R-module M,and K=kerf.Thenfor any G∈F.

Theorem 2.1 Let R be a ring.For the following statements:

(1)R is right F-perfect;

(2)R is semiperfect and F-covers of f i nitely generated R-modules are f i nitely generated;

(3)Finitely generated F right R-modules are projective and F-covers of f i nitely generated right R-modules are f i nitely generated;

(4)F-covers of f i nitely generated right R-modules are projective;

(5)F-covers of cyclic right R-modules are projective, we have(1)?(3)?(4)?(5)and(1)?(2).

Proof.(1)?(2).Let M be a f i nitely generated right R-module and f:F→ M be an F-covers of M.Suppose that g:Rn→M is an epimorphism.Since F is R-projective,by Lemma 2.1,F is Rn-projective.So there exists h:F→Rnsuch that

As f:F→M is a f l at cover of M,there exists k:Rn→F such that

Thus we have the following commutative diagram:

Therefore,

By the de fi nition of an F-cover,kh must be an automorphism of F.Thus k:Rn→F is a split epimorphism.That is,F is a fi nitely generated projective R-module.By Lemma 2.4, f:F→M is a projective of M,and hence R is semiperfect.

(1)?(3).By the proof of(1)?(2),F-covers of fi nitely generated right R-modules are fi nitely generated.Now we show that fi nitely generated F right R-modules are projective. Let M be a fi nitely generated right R-module with M ∈F.Then there exists a projective cover p:P → M with P fi nitely generated.Since R is right F-perfect,any F ∈F is P-projective by Lemma 2.1.That is,for any homomorphism f:F→ M,there exists g:F→P such that

So p:P→M is an F-cover of M,and hence K∈C by Lemma 2.3.That is,

the sequence 0→K→P→M →0 is split,and therefore M is projective.

(3)?(4)and(4)?(5)are clear.

(5)?(1).Let F∈F,I be an ideal of R,π:R→R/I be the natural epimorphism and f:F→R/I be a homomorphism,g:G→R/I be a F-cover of R/I.By hypothesis,G is projective,and hence there is h:G→R such that g=πh.There exists k:F→G such that f=gk by the def i nition of F-cover.

Corollary 2.1([3],Theorem 3.7) For a ring R,the following statements are equivalent:

(a)R is right A-perfect;

(b) R is semiperfect,and f l at covers of f i nitely generated right R-modules are f i nitely generated;

(c) Finitely generated f l at right R-modules are projective,and f l at covers of f i nitely generated right R-modules are f i nitely generated;

(d)Flat covers of f i nitely generated right R-modules are projective;

(e)Flat covers of cyclic right R-modules are projective.

Lemma 2.4 Let f:F→ M be an F-cover of the R-module M.If K ?kerf and F/K∈F,then K=0.

Proof.Suppose that K≤kerf and F/K∈F.Let p:F→F/K be the natural epimorphism.So f induces:F/K→M such that

Since F/K∈F and f:F→M be an F-cover of the R-module M,there exists q:F/K→F with fq= ˉf.That is,we get the following commutative diagram:

Therefore,

Thus qp is an automorphism of F and so

Theorem 2.2 Let R be a ring.Then R is right F-perfect if and only if for any F∈F, and K≤F if F/K is cyclic(f i nitely generated),then F=P⊕Q with Q?K and P is a projective R-module.

Proof. Suppose that R is right F-perfect.Let F be a right R-module with F∈F and K≤F with F/K being cyclic(f i nitely generated).Suppose that g:P→F/K is an F-cover of F/K and f:F→ F/K is the natural epimorphism.Since R is right F-perfect,P is projective,and so there is h:P→F with fh=g.By the def i nition of the F-cover,there exists k:F→P with f=gk,i.e.,we have the commutative diagram:

Thus g=gkh.Therefore,kh is an automorphism of P,and so

Hence imh～=P is projective,and

Conversely,let M be a cyclic(f i nitely generated)R-module and f:F → M be an F-cover of M.Since F/kerf～=M is cyclic(f i nitely generated),by hypothesis,

where Q?K,and P is a projective R-module.So F/Q～=P is projective.By Lemma 2.4, Q=0.Therefore,F=P is projective,and so R is right F-perfect by Theorem 2.1.

Theorem 2.3 A ring R is right F-perfect if and only if for every right R-module F with F∈F;if F=P+U,where P is a f i nitely generated projective summand of F and U≤F, then for some

Proof.Suppose that R is right F-perfect.Let F be a right R-module with F∈F and F=P+U,where P is a f i nitely generated projective summand of F and U≤F.Assume that F=P⊕Q,and p:P→F/U and q:Q→F/U be the canonical mappings.Since Q∈F and R is right F-perfect,by Lemma 2.1,Q is P-projective.So there exists f:Q→P such that

that is,we have the following commutative diagram:

This means that for any x∈Q,

and hence(1-f)(Q)?U.

Now we show that

We have

Thus

and so x=0.

Conversely,let G∈F.We show that G is R-projective,and so R is right F-perfect.

Suppose that p:R→W is an epimorphism and g:G→W is an homomorphism.Let

and

Since p is epimorphism,

Let f:F→R be the projection with respect to the decomposition

Let h=f|G:G→R.Since

for any x∈G,we have

and so

that is,

i.e.,we have the following commutative diagram:

Consequently,R is right F-perfect.

Proposition 2.1 Let F be an R-module with F∈F and f:F→M be an epimorphism. If kerf∈C,then f:F→M is an F-precover of M.

Proof.Since f:F→M is an epimorphism,the sequence

is exact.This induces the exact sequence

for any X∈F.By hypothesis,

and so

is exact.Hence f:F→M is an F-precover of M.

Theorem 2.4 Let R be a ring and I any right ideal of R.Let(F,C)be a cotorsion theory such that if C∈C as an R/I-module,then C∈C as R-module.Then R is right F-perfect if and only if I∈C.

Proof. Let F be a right R-module with F∈F,and I be a right ideal of R.The exact sequence

induces the exact sequence

Since I∈C,

and so HomR(F,R)→HomR(F,R/I)is an epimorphism.Therefore,F is projective relative to R,and hence R is right F-perfect.

Conversely,suppose that R is right F-perfect.Let J=J(R).By Theorem 2.1,F-covers of cyclic right R-modules are projective,and hence F-covers and projective covers of cyclic right R-modules are the same.Since the natural map p:R→R/J is the projective cover of the cyclic right R-module R/J,it is also its F-cover.Thus,by Lemma 2.3,

Furthermore,R/J is a semisimple ring,and so R/J is injective as an R/J-module.By hypothesis,R/J is injective as an R-module,and so R/J∈C as a right R-module.Now consider the exact sequence

Since C is closed under extensions,R∈C.Let I be a proper right ideal of R,and let F∈F. The exact sequence

induces the exact sequence

Since R is right F-perfect,HomR(F,R)→HomR(F,R/I)is an epimorphism.Therefore,

and hence I∈C.

3 F-perfect Modules

In this section,we assume that(F,C)is a perfect cotorsion theory.

De fi nition 3.1 Let M and N be R-modules.Then N is said to be M-cyclic(respectively, fi nitely M-generated)if there is an epimorphism M →N(respectively,Mn→N for some n≥1).

De fi nition 3.2 We call an R-module M F-perfect if F-cover of every M-cyclic R-module is projective.

Proposition 3.1 Let M be an R-module.Then M is F-perfect if and only if every R-module F∈F is M-projective and the F-cover of M is projective.

Proof.Suppose that M is F-perfect.Let F be an R-module with F∈F.We show that F is M-projective.Let p:M→N be an epimorphism,and f:F→N be a homomorphism. Suppose that g:G→N is an F-cover of N.So there is h:F→G with gh=f.Since M is F-perfect,G is projective.Thus there is q:G→M with pq=g.Therefore,

So F is M-projective.It is easy to prove that the F-cover of M is projective.

Conversely,let N be an M-cyclic R-module and f:F→N be an F-cover of N.We want to show that F is projective.Let p:M →N be an epimorphism and g:G→M be an F-cover of M.There is h:G→F such that fh=pg(by the def i nition of F-cover),that is,we have the commutative diagram:

Since every F∈F is M-projective,there exists q:F→M with pq=f.Again by the def i nition of f l at cover,there exists k:F→G with gk=q.Thus

i.e.,we have the commutative diagram:

Therefore,hk is an automorphism of F,and hence F is isomorphic to a summand of G. Since G is projective,F is also projective.Consequently,M is F-perfect.

Corollary 3.1 The class of F-perfect modules is closed under factor modules and extensions.In particular,for modules M1,M2,···,Mn,the sumis F-perfect if and only if each Mis F-perfect.i

Proof. By the def i nition of F-perfect modules and Proposition 3.1 the proof is clear.

Proposition 3.2 An R-module M is F-perfect if and only if for any R-module F∈F and any submodule K of F,if F/K is f i nitely M-generated(or M-cyclic),then F=P⊕Q with P projective and Q?K.

Proof. The proof is similar to that of Theorem 2.2.

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tion:16D50,16D40,16L30

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*Received date:Aug.23,2010.