XIANG YUE-MING

(Department of Mathematics and Applied Mathematics,Huaihua University, Huaihua,Hunan,418000)

Communicated by Du Xian-kun

PS-injective Modules,PS-f l at Modules and PS-coherent Rings

XIANG YUE-MING

(Department of Mathematics and Applied Mathematics,Huaihua University, Huaihua,Hunan,418000)

Communicated by Du Xian-kun

A left ideal I of a ring R is small in case for every proper left ideal K of R, K+I/=R.A ring R is called left PS-coherent if every principally small left ideal Ra is f i nitely presented.We develop,in this paper,PS-coherent rings as a generalization of P-coherent rings and J-coherent rings.To characterize PS-coherent rings,we f i rst introduce PS-injective and PS-f l at modules,and discuss the relation between them over some spacial rings.Some properties of left PS-coherent rings are also studied.

PS-injective module,PS-f l at module,PS-coherent ring

1 Introduction

Throughout this paper,R is an associative ring with identity and all modules are unitary. The Jacobson radical of R is denoted by J(R)and its right singular ideal is denoted by Z(RR).Let M and N be R-modules.Hom(M,N)(resp.Extn(M,N))means HomR(M,N) (resp.ExtnR(M,N)),and similarly M⊗N(resp.Torn(M,N))denotes M⊗RN(resp. TorRn(M,N)).The character module M+is def i ned by M+=HomZ(M,Q/Z).If X is a subset of R,the right(resp.left)annihilator of X in R is denoted by rR(X)(resp.lR(X)). For the usual notations we refer the reader to[1–4].

Let R be a ring.A left R-module M is called f i nitely presented if there is an exact sequence 0→K→F→M→0,where F is f i nitely generated free and K is f i nitely generated.A ring R is called left coherent if every f i nitely generated left ideal of R is f i nitely presented.Coherent rings and their generalizations have been extensively studied by many authors(see[5–8]).A left ideal I of a ring R is small in case for every proper left ideal K of R,K+I/=R.A ring R is said to be left J-coherent(see[5])(resp.P-coherent(see[8]))if every f i nitely generated small(resp.principally)left ideal of R is f i nitely presented.In Section 3 of this paper,we say that a ring R is left PS-coherent if every principally small left ideal of R is f i nitely presented.The concept of PS-coherent rings is introduced as a proper generalization of J-coherent rings and P-coherent rings.Some examples of PS-coherent rings are given,and some properties of PS-coherent rings are studied.It is shown that R is left J-coherent if and only if the matrix ring Mn(R)is left PS-coherent for all n≥1.If R is semiregular,then R is left P-coherent if and only if R is left PS-coherent.

In order to characterize PS-coherent rings,PS-f l at modules are f i rstly introduced in Section 2.We discuss the relation between PS-injective modules and PS-f l at modules over some spacial rings.In light of these facts,we give some characterizations of semiprimitive rings(that is,J(R)=0).

Let C be the class of R-modules.For an R-module M,a homomorphism g:C→M with C∈C is called a C-cover(see[2])of M if the following hold:

(1)For any homomorphism g′:C′→M with C′∈C,there exists a homomorphism f:C′→C with g′=gf;

(2)If f is an endomorphism of C with gf=g,then f must be an automorphism.

If(1)holds but(2)may not,g:C→M is called a C-precover.Dually,we have the def i nition of a C-(pre)envelope.C-covers and C-envelopes may not exist in general,but if they exist,they are unique up to isomorphism.In Section 3 of this paper,we show that R is left PS-coherent if and only if every right R-module has a PS-f l at preenvelope.If R is left PS-coherent,then every left R-module has a PS-injective cover.Furthermore,we consider when every left R-module has an epimorphic PS-injective cover and when every right R-module has a monomorphic PS-f l at preenvelope.

2 PS-injective Modules and PS-f l at Modules

We start with the following def i nition.

Def i nition 2.1Let R be a ring and Ra be any principally small left ideal.A left R-module M is called PS-injective if every R-homomorphism f:Ra→M can be extended to R→M,equivalently,if Ext1(R/Ra,M)=0.A right R-module N is said to be PS-f l at if the sequence 0→N⊗Ra→N⊗R is exact,or equivalently,Tor1(N,R/Ra)=0.Similarly, we have the concept of right PS-injective modules and left PS-f l at modules.

Remark 2.1(1)A left R-module M is said to be divisible(see[8])(or P-injective)if Ext1(R/Ra,M)=0 for all a∈R.A right R-module N is called torsionfree if

Clearly,every divisible module(resp.torsionfree module)is PS-injective(resp.PS-f l at). But the converse is not true in general.For example,let R=Z be the ring of integers.Then every module is PS-injective and PS-f l at because J(R)=0.However,Z is not a divisible Z-module and Z2is not a torsionfree Z-module.

(2)A right R-module N is PS-f l at if and only if N+is PS-injective by the standard isomorphism

for any principally small left ideal Ra.

(3)In what follows,we denote the class of left PS-injective modules(resp.right PS-f l at modules)by PSI(resp.PSF).It is easy to verify that PSI is closed under extensions, direct products,direct sums and direct summands,and PSF is closed under extensions, direct limits,direct sums and direct summands.

Lemma 2.1The classes PSI and PSF are closed under pure submodules.

Proof.Let N be a pure submodule of a left PS-injective R-module M.For any principally small left ideal Ra,there is an exact sequence:

Note that Ext1(R/Ra,M)=0.But R/Ra is f i nitely presented and N is a pure submodule, so π is epic.Thus,

Therefore,N is PS-injective.

Now let A be a pure submodule of a right PS-f l at R-module B.Then we have a split exact sequence:

By Remark 2.1(2),B+is PS-injective,so is A+by Remark 2.1(3),which implies that A is PS-f l at.

Theorem 2.1Let R be a ring and I be a ideal of R such that R/I is semisimple ring. Then R/I is right(resp.left)PS-f l at if and only if R/I is left(resp.right)PS-injective. Proof.It suffices to give a proof for the one case.The other case is similar.

Sufficiency.Assume that R/I is right PS-f l at.For R/Ra with a∈J(R),there is a standard isomorphism

since Q/Z is an injective Z-module.On the other hand,(R/I)+I=0,so(R/I)+is a semisimple right R-module,and

where Sαis a simple right R-module.By Def i nition 2.1,

Then

Since Sαis the summand of R/I,we have

Hence,

So I(R/I)++=0,and then(R/I)++is a semisimple R-module.Thus R/I is the summand of(R/I)++since R/I is a pure submodule of(R/I)++.Hence,Ext1(R/Ra,R/I)=0. Therefore,R/I is left PS-injective.

Necessity.Assume that R/I is left PS-injective.For any a∈J(R),

where R/I is a right R-module.Since(R/I)+is semisimple,

with Sαa simple left R-module.By Def i nition 2.1,Ext1(R/Ra,R/I)=0.Hence,

So

Thus Tor1(R/I,R/Ra)=0.Therefore,R/I is right PS-f l at.

Recall that a ring R is said to be semilocal if R/J(R)is semisimple.So we have the following corollary.

Corollary 2.1Let R be a semilocal ring.Then R/J(R)is right(resp.left)PS-f l at if and only if R/J(R)is left(resp.right)PS-injective.

Corollary 2.2Let R be a commutative ring.Then every simple R-module S is PS-injective if and only if S is PS-f l at.

Theorem 2.2The following statements are equivalent for a ring R:

(1)R is semiprimitive;

(2)Every right(or left)R-module is PS-injective;

(3)Every right(or left)simple R-module is PS-injective;

(4)Every right(or left)principle small ideal is PS-injective;

(5)Every left(or right)R-module is PS-f l at.

Moreover,if R is semilocal,then the above conditions are equivalent to

(6)Every left(or right)simple R-module is PS-f l at.

Proof.(1)⇔(2)⇔(3)⇔(4)follow from Proposition 2.18 of[9].

(2)⇒(5).Let M be any left(right)R-module.By hypothesis,M+is right(left)PS-injective.Then M is left(right)PS-f l at by Remark 2.1(2).

(5)⇒(2).Note that R/Ra is f l at for any a∈J(R).So R/Ra is projective because R/Ra is also f i nitely presented.Therefore,every left R-module is PS-injective.

(5)⇒(6)is clear.

(6)⇒(3)follows from Theorem 2.1.

This completes the proof.

A ring R is called a left(right)V-ring(see[10])if every simple left(right)R-module is injective.A ring R is called a left(right)SF-ring(see[11])if every simple left(right) R-module is f l at.Now we have some results as corollaries.

Corollary 2.3Every left(or right)V-ring is semiprimitive.

Corollary 2.4Every semilocal,left(or right)SF-ring is semiprimitive.

Example 2.1There exists a ring whose every simple singular right R-module is PS-injective with J(R)/=0.For example,let,where F is a f i eld.Then J(R)/=0.Note thatis the unique maximal essential right ideal of R.It is easy to show that every simple singular right R-module is PS-injective.

In general,if R is a ring whose every simple singular right R-module is P-injective, then J∩Z(RR)=0(see Proposition 3 of[12]).Now we extend this result to PS-injective modules.

Lemma 2.2If R is a ring whose every simple singular right R-module is PS-injective, then J∩Z(RR)=0.

Proof.Take any b∈J∩Z(RR).If b/=0,then rR(b)+RbR is an essential right ideal of R.If rR(b)+RbR/=R,there exists a maximal essential right ideal T of R such that rR(b)+RbR⊆T.By hypothesis,R/T is PS-injective.Note that rR(b)⊆T.Then the R-homomorphism f:bR→R/T by br→r+T is well def i ned.So f=(c+T)for some c∈R.Then f(b)=1+T=cb+T.Note that cb∈RbR⊆T,so 1∈T,a contradiction. This proves that rR(b)+RbR=R,and hence rR(b)=R because RbR is a small ideal of R. This implies b=0,which is a required contradiction.

Theorem 2.3The following statements are equivalent for a ring R:

(1)R is semiprimitive;

(2)R is right PS-injective and every simple singular right R-module is PS-injective. Proof.(1)⇒(2)is clear.

(2)⇒(1).Since R is right PS-injective,by Theorem 2.6 of[9],J⊆Z(RR).Then J=J∩Z(RR)=0 by Lemma 2.2,as desired.

3 PS-coherent Rings

Def i nition 3.1A ring R is called left PS-coherent provided that every principally small left ideal is f i nitely presented,or equivalently,every principally left ideal in J(R)is f i nitely presented.Similarly,we have the concept of right PS-coherent rings.

Remark 3.1Here are some examples of PS-coherent rings.

(1)Obviously,left P-coherent rings are left PS-coherent.In particular,a domain is left and right PS-coherent.

(2)Left J-coherent rings are left PS-coherent.Moreover,semiprimitive rings are left and right PS-coherent.

Example 3.1Let R be a valuation ring of rankR＞1.Then R[[x]],the ring of power series in one variable x,is a commutative domain,and so it is PS-coherent.But R[[x]]is not a J-coherent ring by Example 3.16 of[5].

Left PS-coherent rings are left min-coherent,that is,every minimal left ideal is f i nitely presented(see[7]).In fact,in view of Lemma 10.22 of[4],every minimal left ideal of R is either nilpotent or a direct summand of R.

It has been shown that R is a left coherent ring if and only if Mn(R)is a left P-coherent ring for each n≥1(see Proposition 2.4 of[8]).Analogously,we have the following result.

Proposition 3.1The following statements are equivalent for a ring R:

(1)R is left J-coherent;

(2)Mn(R)is left PS-coherent for all integers n≥1.

Proof.(1)⇒(2)follows from Corollary 2.11 of[5]and Remark 3.1(2).

(2)⇒(1).Write S=Mn(R).Let I=Ra1+Ra2+···+Ranbe a f i nitely generated left ideal in J(R).Put

Then

By hypothesis,S¯a is a f i nitely presented S-module.Then there exists a left S-module exact sequencewhere K is a f i nitely generated left S-module.Note that S is also a f i nitely generated free left R-module and K is a f i nitely generated left R-module.Then S¯a is a f i nitely presented left R-module.So I is f i nitely presented because S¯a~=In.Therefore,R is left J-coherent. This completes the proof.

From the preceding proposition and Example 3.1,being a left PS-coherent ring is not Morita invariant.

Example 3.2Let R=K[x2,x3,yi,xyi]with K a f i eld and x,y1,y2,···indeterminates over K,and G be a free abelian group with rank G=∞.Write the group ring S=RG. By Example 2.8 of[5],S is J-coherent but not coherent.Then the matrix ring Mn(S)is PS-coherent but not P-coherent by of Proposition 3.1 and Proposition 2.4 of[8].

A ring R is called semiregular if R/J(R)is von Neumann regular and idempotents lift modulo J(R),or equivalently if,for any a∈R,there exists e2=e∈Ra such that a(1-e)∈J(R)(see Lemma B.40 of[13]).

Proposition 3.2If R is a semiregular ring.Then R is left P-coherent if and only if R is left PS-coherent.

Proof.Sufficiency is clear.

Necessity.Let Ra be any principally left ideal of R.Since R is semiregular,Ra=Re⊕Rb where e2=e∈R and b∈J(R).Note that Rb is f i nitely presented because R is left PS-coherent.Then Ra is f i nitely presented.Therefore,R is left P-coherent.

Using the analogous argument of Theorem 2.7 of[8]or Theorem 2.13 of[5],we can prove the following theorem.

Theorem 3.1Let R be a ring.Then the following statements are equivalent:

(1)R is a left PS-coherent ring;

(2)Any direct product of copies of RRis PS-f l at;

(3)Any direct product of PS-f l at right R-modules is PS-f l at;

(4)Any direct limit of PS-injective left R-modules is PS-injective;

(5)There is an isomorphism

for any principally small left ideal Ra and any family{Mi}of right R-modules;

(6)A left R-module M is PS-injective if and only if M+is PS-f l at;

(7)A left R-module M is PS-injective if and only if M++is PS-injective;

(8)A right R-module N is PS-f l at if and only if N++is PS-f l at;

(9)lR(a)is f i nitely generated for any a∈J(R);

(10)Any right R-module has a PS-f l at preenvelope.

The below theorem is a characterization of commutative PS-coherent rings.

Theorem 3.2The following statements are equivalent for a commutative ring R:

(1)R is a PS-coherent ring;

(2)For any PS-injective R-module M and injective R-module N,the homomorphic module Hom(M,N)is PS-f l at;

(3)The homomorphic module Hom(R+,N)is PS-f l at for any injective R-module N.

Proof.(1)⇒(2).For any PS-injective R-module M and injective R-module N,note that Q/Z is injective cogenerator.Then there exists an exact sequence 0→N→∏(Q/Z).Since N is injective,∏(Q/Z)~=N⊕C.Then

But

So Hom(M,N)is a summand of∏M+.In addition,by Theorem 3.1,∏M+is PS-f l at. Therefore,Hom(M,N)is PS-f l at by Remark 2.1(3).

(2)⇒(3)is clear since R+is(PS-)injective.

(3)⇒(1).Since R is a pure submodule of R++,∏R is a pure submodule of∏R++by Lemma 1(2)of[14].Note that

Thus∏R++is PS-f l at by hypothesis,and henceR is PS-f l at.So R is PS-coherent by Theorem 3.1 again.

This completes the proof.

The following example shows that the def i nition of a PS-coherent ring is not left-right symmetric.

Example 3.3Let L=Q(x2,x3,···)be a subf i eld of K=Q(x1,x2,···)with Q the f i eld of rational numbers,and there exists a f i eld isomorphism φ:K→L.We def i ne a ring by taking R=K×K with multiplication

It is easy to see that R has exactly three right ideals:(0),R,and(0,K)=(0,1)R.So R is right PS-coherent.Let a=(0,1)∈J(R).Then lR(a)is not f i nitely generated.By Theorem 3.1,R is not left PS-coherent.

Corollary 3.1Let R be a left PS-coherent ring.Then every left R-module has a PS-injective cover.

Proof.Let 0→A→B→C→0 be a pure exact sequence of left R-modules with B∈PSI.Then 0→C+→B+→A+→0 splits.Since B+is PS-f l at by Theorem 3.1, C+is PS-f l at,and hence C is PS-injective.By Theorem 2.5 of[15],every left R-module has a PS-injective cover.

Now we consider when every left R-module has an epimorphic PS-injective cover and when every right R-module has a monomorphic PS-f l at preenvelope.

Proposition 3.3Let R be a left PS-coherent ring.Then the following statements are equivalent:

(1)R is left PS-injective;

(2)Every left R-module has an epimorphic PS-injective cover;

(3)Every right R-module has a monomorphic PS-f l at preenvelope;

(4)Every injective right R-module is PS-f l at;

(5)Every f l at left R-module is PS-injective.

Proof.(1)⇒(2).Since R is left PS-coherent,every left R-module has a PS-injective cover g:D→M by Corollary 3.1.In addition,for any left R-module M,there is an epimorphism f:R(I)→M.RR is PS-injective,so is R(I)by Remark 2.1(3).Thus there is a homomorphism h:R(I)→D such that gh=f.Since f is epimorphic,so is g.

(2)⇒(1).By hypothesis,R has an epimorphic PS-injective cover φ:D→R.Then we have an exact sequence 0→K→Dφ-→R→0 with K=kerφ and D PS-injective.Since R is f l at,the exact sequence is pure,so K is a pure submodule of D.Thus R is PS-injective as a left R-module by the proof of Corollary 3.1.

(1)⇒(3).Let M be any right R-module.Then M has a PS-f l at preenvelope f:M→F by Theorem 3.1.Sinceis a cogenerator in the category of right R-modules,there is an exact sequence.By Theorem 3.1,is PS-f l at.So there exists a homomorphism g:such that gf=i.Since i is a monomorphism,so is f.

(3)⇒(4).Note that the FP-injective right R-module E embeds in a PS-f l at right R-module by(3).Thus E is PS-f l at by Lemma 2.1.

(4)⇒(5).For any f l at left R-module F,is injective.Thenis PS-f l at by hypothesis,and hence F is PS-injective by Theorem 3.1.

(5)⇒(1)is trivial.

Example 3.4Let Rbe a triangular ring,then R is left hereditary.Let,and x∈Q.Then 0/=a∈J(R).So R is not a semiprimitive ring.We claim that R is not left PS-injective.

Proof.We f i rst prove that every quotient module of a PS-injective left R-module is PS-injective.Let B be any PS-injective left R-module and A⊆B.For any principally small left ideal Ra of R and a homomorphism f:Ra→B/A,since R is left hereditary,Ra is projective.So there is a homomorphism g:Ra→B such that πg=f,where π:B→B/A is the nature epimorphism.Then there is a homomorphism h:R→B such that hi=g since B is PS-injective,where i:Ra→R is an inclusion.Thus,f=πhi,and hence B/A is PS-injective.For any left R-module M,by Proposition 4 of[16],there is a monomorphic PS-injective cover α:I→M.Suppose that R is left PS-injective.Then α is epimorphic by Proposition 3.3,and hence M is PS-injective.By Theorem 2.2,R is semiprimitive,a contradiction.

Remark 3.2By the preceding example,there exists a ring whose every left R-module has a PS-injective cover that may not be an epimorphism.

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tion:16P70,16N20,16D10

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1674-5647(2013)02-0121-10

Received date:Dec.19,2010.

E-mail address:xymls999@126.com(Xiang Y M).