Weakly I-semiregular Rings and I-semiregular Rings

Zhanmin Zhu

Department of Mathematics,Jiaxing University,Jiaxing 314001,China.

Abstract.Let I be an ideal of a ring R.We call R weakly I-semiregular if R/I is a von Neumann regular ring.This definition generalizes I-semiregular rings.We give a series of characterizations and properties of this class of rings.Moreover,we also give some properties of I-semiregular rings.

Key words:Weakly I-semiregular rings,I-semiregular rings,n-injective modules,n-flat modules,(m,n)-injective modules.

1 Introduction

Throughout this paper,m,n are positive integers,R is an associative ring with identity,I is an ideal of R,J=J(R)is the Jacobson radical of R and all modules considered are unitary.

Recall that a ring R is called semiregular[11],if for any a∈R,there exists e2=e∈aR such that(1−e)a∈J.By[11,Theorem 2.9],a ring R is semiregular if and only if R/J is von Neumann regular and idempotents can be lifted modulo J.In[12],Nicholson and Yousif extend the concept of semiregular rings to I-semiregular rings.Let I be an ideal of R.Then following[12],an element a∈R is called left I-semiregular if there exists e2=e∈Ra such that a(1−e)∈I,equivalentely if there exists f2=f∈aR such that(1−f)a∈I;a ring R is called left I-semiregular if every element of R is I-semiregular.It is easy to see that I-semiregular rings are left-right symmetric.I-semiregular rings have been studied by many authors(see,for example[2,12,13,16,17,20]).By[12,Theorem 1.2]or[13,Theorem 28],we see that if R is left I-semiregular,then R/I is regular and idempotents can be lifted modulo I.

In this article,we extend the concept of I-semiregular rings to weakly I-semiregular rings.Let I be an ideal of R.We will call R weakly I-semiregular if R/I is regular.A series of characterizations and properties of this class of rings will be given,and some properties of I-semiregular rings will be given too.

For any module M,M+denotes HomZ(M,Q/Z),where Q is the set of rational numbers,and Z is the set of integers.In general,for a set S,we write Sm×nfor the set of all formal m×n matrices whose entries are elements of S,and Sn(resp.,Sn)for the set of all formal n×1(resp.,1×n)matrices whose entries are elements of S.Let M be a left R-module,X⊆Mnand A⊆Rm×n.Then we define rMn(A)={u∈Mn:au=0,∀a∈A},and lRm×n(X)={a∈Rm×n:ax=0,∀x∈X}.

2 Weakly I-semiregular rings

At the begin of this section,we introduce the concept of weakly I-semiregular rings as following.

Definition 2.1.Let I be an ideal of a ring R.Then R is said to be weakly I-semiregular if R/I is a von Neumann regular ring.

Let T be an ideal of a ring R.Then following[13],we say that idempotents lift strongly modulo T,if a2−a∈T,then there exists e2=e∈aRa such that e−a∈T.

Example 2.1.(1)By[13,Theorem 28],a ring R is I-semiregular if and only if it is weakly I-semiregular and idempotents lift strongly modulo I.In particular,a ring R is a semiregular ring and only if it is a weakly J(R)-semiregular and idempotents lift strongly modulo J(R).

(2)By[17,Theorem 1.6],a ring R is Sr-semiregular if and only if it is weakly Srsemiregular,where Sr=Soc(RR).

(3)By[13,Example 24],there exists a commutative ring R which contains an ideal I such that R/I～=Z6,but idempotents do not all lift modulo I.Note that the ring Z6is von Neumann regular,so R is weakly I-semiregular but it is not I-semiregular.Also,Z is weakly I-semiregular for each non-zero semiprimitive ideal I,but not I-semiregular.

Let M be an R-module and N a submodule of M.According to[25],N is said to have a weak supplement L in M if N+L=M and N∩L≪M,and M is called weakly supplemented if every submodule N of M has a weak supplement.It is easy to see thatRR is weakly supplemented if and only if for any left ideal L of R,there is a left ideal K such that L+K=R and L∩K⊆J(R).Inspired by this result,we have the following definition.

Definition 2.2.Let I be an ideal of a ring R.Then a left ideal L of R is said to be I-weak supplemented inRR if there exists a left ideal K such that L+K=R and L∩K⊆I.In this case,we call K an I-weak supplement of L in R.

Similarly,we can define the concept of I-weak supplement of a right ideal.

Example 2.2.Let L be a left ideal of R.Then a left ideal K is a J(R)-weak supplement of L in R if and only if K is a weak supplement of L in R.

Theorem 2.1.Let I be an ideal of a ring R.Then the following statements are equivalent:

(1)R is a weakly I-semiregular ring.

(2)Every finitely generated left ideal of R has an I-weak supplement inRR.

(3)Every principal left ideal of R has an I-weak supplement inRR.

(4)Every finitely generated right ideal of R has an I-weak supplement in RR.

(5)Every principal right ideal of R has an I-weak supplement in RR.

Proof.(1)⇒(2).Let L be a finitely generated left ideal of R.For any X⊆R,set X={x+I:x∈X}.Since R is regular,we have R=L⊕H,where H is a left ideal of R.Let K=H+I.Then it is easy to see that K is a left ideal,R=L+K,and L∩K⊆I,i.e.,K is an I-weak supplement of L.

(2)⇒(3).It is obvious.

(3)⇒(1).Let a∈ R.By assumption there exists an I-weak supplement K of Ra.Let 1=ba+k,where b∈R,k∈K.Then k=1−ba,and so,by[9,Lemma 3.4],Rak=Ra∩Rk⊆Ra∩K⊆I.It follows that a−aba=ak∈I,and thus R/I is von Neumann regular.

(1)⇒(4)⇒(5)⇒(1)analogous.

Corollary 2.2.([9,Proposition 3.18]and[14,Proposition 2.3]).The following statements are equivalent for a ring R:

(1)R/J(R)is von Neumann regular.

(2)Every finitely generated left ideal of R has a weak supplement inRR.

(3)Every principal left ideal of R has a weak supplement inRR.

(4)Every finitely generated right ideal of R has a weak supplement in RR.

(5)Every principal right ideal of R has a weak supplement in RR.

Recall that an element a∈R is called regular if a=aba for some b∈R,and we call an ideal I of R regular if every element in I is regular.The following proposition improves[23,Theorem 6.3].

Proposition 2.1.Let I be an ideal of a ring R.Then R is von Neumann regular if and only if R is weakly I-semiregular and I is regular.

Proof.We need only to prove the sufficiency.For any a∈R,since R is weakly I-semiregular,there exists b∈R such that a−aba∈I.But I is regular,a−aba=(a−aba)c(a−aba)for some c∈R,so a=a(b+c−cab−bac+bacab)a.And hence R is von Neumann regular.

Recall that a left R-module M is called n-injective[1,10],if every R-homomorphism from an n-generated left ideal to M extends to a homomorphism of R to M;a left R-module M is called F-injective[7]if M is n-injective for every positive integer n;a left R-module M is called P-injective[10]if it is 1-injective.In[23],the three concepts are extended to I-n-injective modules,I-F-injective modules and I-P-injective modules respectively.Following[23],a left R-module M is called I-n-injective,if every R-homomorphism from an n-generated left ideal in I to M extends to a homomorphism of R to M;a left R-module M is called I-F-injective if M is I-n-injective for every positive integer n;a left R-module M is called I-P-injective if it is I-1-injective.We recall also that a right R-module M is said to be n-flat[1,6],if for every n-generated left ideal T,the canonical map M⊗T→M⊗R is monic;MRis said to be P-flat[4]if it is 1-flat.It is easy to see that MRis flat if and only if it is n-flat for every positive integer n.In[23],the three concepts are extended to I-n-flat modules,I-P-flat modules and I-flat modules respectively.Following[23],a right R-module M is said to be I-n-flat,if for every n-generated left ideal T in I,the canonical map M⊗T→M⊗R is monic;MRis said to be I-P-flat if it is I-1-flat;MRis said to be I-flat if it is I-n-flat for every positive integer n.We note that there is another definition of I-flat modules in[24],we don’t know whether the two definitions coincide.In this paper,we will always use the definition in[23].The following theorem improves[14,Theorem 2.11],[19,Theorem 2.11(2)]and[23,Theorem 2.2,Corollary 3.2(1)].

Theorem 2.3.Let I be an ideal of a ring R.If R is weakly I-semiregular and n is a positive integer.Then:

(1)A left R-module M is n-injective if and only if M is I-n-injective.

(2)A right R-module M is n-flat if and only if M is I-n-flat.

Proof.(1)We need only to prove the sufficiency.Let L=Ra1+Ra2+···+Ranbe an ngenerated left ideal of R and f a homomorphism from L to M.Since R is weakly I-semiregular,by Theorem 2.1,L has an I-weak supplement in R.That is,there exists a left ideal K of R such that L+K=R and L∩K ⊆ I.And so,there exist r1,r2,···,rn∈ R and k∈K such that r1a1+r2a2+···+rnan+k=1 and L∩Rk⊆ L∩K⊆ I.By Lemma 2.1,is an n-generated left ideal in I.Since M is I-n-injective,there exists a homomorphism g ofRR to M such that g|L∩Rk=f|L∩Rk.Note that L+Rk=R.So,for any x∈R,there exist x1∈L and x2∈Rk such that x=x1+x2.Now we define φ :RR →RR by φ(x)=f(x1)+g(x2),then it is easy to check that φ is well-defined and φ extends f.

(2)By[21,Theorem 4.3]and[23,Theorem 3.1]and(1),we have that M is n-flat⇔M+is n-injective⇔M+is I-n-injective⇔M is I-n-flat.

Corollary 2.4.Let I be an ideal of a ring R.If R is weakly I-semiregular,then:

(1)A left R-module M is P-injective if and only if M is I-P-injective.

(2)A left R-module M is F-injective if and only if M is I-F-injective.

(3)a right R-module M is P-flat if and only if M is I-P-flat.

(4)a right R-module M is flat if and only if M is I-flat.

Recall that a left R-module M is called JP-injective[18],if every R-homomorphism from a principal left ideal in J(R)to M extends to a homomorphism of R to M;a left R-module M is called J-injective[5],if every R-homomorphism from a finitely generated left ideal in J(R)to M extends to a homomorphism of R to M;The following corollary improves[14,Corollary 2.12]and[23,Corollary 2.3].

Corollary 2.5.Let R be a ring.If R/J(R)is regular,then:

(1)A left R-module M is P-injective if and only if M is JP-injective.

(2)A left R-module M is F-injective if and only if M is J-injective.

Corollary 2.6.If R/J(R)is regular,Then R is left F-injective if and only if R is left J-injective.

Recall that R is called left coherent[15]if every finitely generated left ideal is finitely presented;R is called left n-coherent[1]if every n-generated left ideal is finitely presented;R is called left semihereditary if every finitely generated left ideal is projective;R is called left n-semihereditary[22]if every n-generated left ideal is projective.Let R be a ring and I be an ideal of R.R is called left I-n-coherent[23]if every n-generated left ideal in I is finitely presented;R is called left I-n-semihereditary[23]if every n-generated left ideal in I is projective.

Theorem 2.7.Let I be an ideal of a ring R.If R is weakly I-semiregular,then:

(1)R is left n-coherent if and only if it is left I-n-coherent.

(2)R is left n-semihereditary if and only if it is left I-n-semihereditary.

Proof.(1)It follows from Theorem 2.3(2)and[23,Theorem 4.2(4),Corollary 4.1(4)].

(2)It follows from Theorem 2.3(1)and[23,Theorem 5.1(6),Corollary 5.1(6)].

Recall that R is called left I-coherent[24]if every finitely generated left ideal in I is finitely presented;R is called left I-P-coherent[23]if it is left I-1-coherent;R is called left I-semihereditary[23]if every finitely generated left ideal in I is projective;R is called left PP[8]if every principal left ideal is projective;R is called left I-PP[23]if every principal left ideal in I is projective.

Corollary 2.8.Let I be an ideal of a ring R.If R is weakly I-semiregular,then:

(1)R is left coherent if and only if it is left I-coherent.

(2)R is left P-coherent if and only if it is left I-P-coherent.

(3)R is left semihereditary if and only if it is left I-semihereditary.

(4)R is left PP if and only if it is left I-PP.

Definition 2.3.Let n be a positive integer and I be an ideal of R.A left R-module M is called simple I-n-injective if,for every n-generated left ideal L⊆I,every R-linear map γ:L→M with γ(L)simple extends to a homomorphism of R to M.A left R-module M is called simple n-injective if it is simple R-n-injective.

Theorem 2.9.Let I be an ideal of a ring R.If R is weakly I-semiregular and n is a positive integer.Then a left R-module M is simple n-injective if and only if M is simple I-n-injective.

Proof.It is similar to the proof of Theorem 2.3.

About I-n-injective rings,we have the following result.

Theorem 2.10.Let I be an ideal of a ring R.Then the following statements are equivalent:

(1)R is left I-n-injective.

(2)Every homomorphism from an n-generated left ideal contained in I to R with principal image can be extended to a homomorphism from R to R.

Proof.(1)⇒(2).It is clear.

(2)⇒(1).Assume(2).Then it is clear that R is left I-P-injective.Now let K=Rb be a cyclic left ideal in I and L an(n−1)-generated left ideal in I.Let a∈rR(K∩L)and define f:K+L→R by f(k+l)=ka for all k∈K and l∈L.Then it is easy to see that R is a left R-homomorphism and Im(f)=Rba is principal.By hypothesis,f can be extended to a homomorphism from R to R,and so f= ·c for some c∈ R.Thus,(k+l)c=ka for all k∈K and l∈L.Let l=0.Then a−c∈rR(K).Let k=0.Then c∈rR(L).Therefore,a=(a−c)+c∈rR(K)+rR(L),it follows that rR(K∩L)⊆rR(K)+rR(L).The inverse inclusion is clear.So rR(K∩L)=rR(K)+rR(L).By[23,Theorem 2.1(6)],R is left I−n-injective.

Corollary 2.11.The following statements are equivalent for a ring R:

(1)R is left n-injective.

(2)Every homomorphism from an n-generated left ideal to R with principal image can be extended to a homomorphism from R to R.

Theorem 2.12.Let I be an ideal of a ring R.If R is weakly I-semiregular and n is a positive integer,then the full matrix ring Mn(R)is weakly Mn(I)-semiregular.

Proof.It is easy to see that Mn(R)/Mn(I)～=Mn(R/I).Since R/I is von Neumann regular and the full matrix ring of a von Neumann regular ring is von Neumann regular,so Mn(R)/Mn(I)is von Neumann regular,as required.

3 I-semiregular rings

Following[2],an element m of a module M is called I-semiregular if there is a decomposition M=P⊕Q,where P is projective,P⊆Rm and Rm∩Q⊆IM,and M is called I-semiregular if every element of M is I-semiregular.By[2,Theorem 2.2],a module M is I-semiregular if and only if for every finitely generated submodule N of M,there exists a decomposition M=P⊕Q,where P is projective,P⊆N and N∩Q⊆IM.And by[2,Theorem 2.6],direct sums and direct summands of I-semiregular modules are I-semiregular.By[12,Lemma 1.1],it is easy to see that a ring R is I-semiregular if and only ifRR is I-semiregular.We recall also that a left R-module M is called(m,n)-injective[3]if every R-homomorphism from an n-generated submodule T of Rmto M extends to a homomorphism of Rmto M;a left R-module M is called FP-injective[15]if M is(m,n)-injective for every pair of positive integers m,n;a left R-module M is called I-(m,n)-injective[24]if every R-homomorphism from an n-generated submodule T of Imto M extends to a homomorphism of Rmto M;a left R-module M is called I-FP-injective[24]if M is I-(m,n)-injective for every pair of positive integers m,n;a right R-module B is said to be(m,n)-flat[21]if for every n-generated submodule T in Rm,the canonical map B⊗T→B⊗Rmis monic;a right R-module B is said to be I-(m,n)-flat if for every n-generated submodule T in Im,the canonical map B⊗T→B⊗Rmis monic.

Lemma 3.1.[24,Lemma 2.9].For a right R-module B,the following statements are equivalent:

(1)B is I-(m,n)-flat.

(2)B+is I-(m,n)-injective.

Theorem 3.1.Let I be an ideal of a ring R.If R is I-semiregular and m,n are two positive integers,then:

(1)A left R-module M is(m,n)-injective if and only if M is I-(m,n)-injective.

(2)A right R-module M is(m,n)-flat if and only if M is I-(m,n)-flat.

Proof.(1)Necessity is clear.To prove sufficiency,let N be an n-generated submodule of the left R-module Imand f:N→M be a left R-homomorphism.Since R is I-semiregular,by[2,Theorem 2.6],Rmis I-semiregular.So,by[2,Theorem 2.2],Rm=P⊕K,where P is projective,P⊆N and N∩K⊆IRm⊆Im.Hence Rm=N+K,N=P⊕(N∩K),and so N∩K is n-generated.Since M is I-(m,n)-injective,there exists a homomorphism g:Rm→M such that g(x)=f(x)for all x∈N∩K.Now let h:,where x=n+k,n∈N,k∈K.Then h is a well-defined left R-homomorphism and h extends f.

(2)We need only to prove the sufficiency.Let M be I-(m,n)-flat.Then by Lemma 3.1,M+is I-(m,n)-injective.So,by(1),M+is(m,n)-injective,and hence M is(m,n)-flat by[21,Theorem 4.3].

Corollary 3.2.Let I be an ideal of a ring R.If R is I-semiregular,then a left R-module M is FP-injective if and only if M is I-FP-injective.In particular,if R is semiregular,then a left R-module M is FP-injective if and only if M is J-FP-injective.

Lemma 3.2.[24,Theorem 3.2].Let M be a left R-module.Then the following statements are equivalent:

(1)M is I-(m,n)-injective.

(2)If x=(m1,m2,...,mn)′∈Mnand A∈In×msatisfy lRn(A)⊆lRn(x),then x=Ay for some y∈Mm.

Theorem 3.3.Let I be an ideal of a ring R and n a positive integer.Then a left R-module M is I-(n,n)-injective if and only if the Mn(R)-module Mn×ndefined by M is Mn(I)-P-injective.

Proof. ⇒.Let X∈Mn×n,A∈Mn(I)with lMn(R)(A)⊆lMn(R)(X).Write X=(X1,X2,···,Xn),where each Xi∈Mn.Then lMn(R)(A)⊆lMn(R)(Xi),i=1,2,···,n.If b∈lRn(A),then by taking B∈Mn(R)such that its first row is b and the other rows are 0’s,we have B∈lMn(R)(A),so BX=0,and thus b∈ lRn(Xi).It shows that lRn(A)⊆ lRn(Xi),i=1,2,···,n.Since M is I-(n,n)-injective,by Lemma 3.2,we have Xi=AYi,where Yi∈ Mn,i=1,2,···,n.Now let Y=(Y1,Y2,···Yn).Then Y ∈ Mn×nand X=AY.Therefore the Mn(R)-module Mn×ndefined by M is Mn(I)-P-injective by[23,Theorem 6.1(4)].

⇐.Let x∈Mnand A∈Mn(I)with lRn(A)⊆lRn(x).Take C∈Mn×nsuch that its first column is x and the other columns are 0’s.If D ∈ lMn(R)(A),write the ith row vectors of D by di,then di∈ lRn(A),and so dix=0,i=1,2,···,n.Thus,DC=0,and then lMn(R)(A)⊆lMn(R)(C).Since the left Mn(R)-module Mn×nis Mn(I)-P-injective,by[23,Theorem 6.1(4)],C=AZ for some Z∈Mn×n.Let z1be the first column of Z.Then x=Az1.So,by Lemma 3.2,M is I-(n,n)-injective.

Corollary 3.4.Let I be an ideal of a ring R.Then a left R-module M is I-FP-injective if and only if the Mn(R)-module Mn×ndefined by M is Mn(I)-P-injective for each positive integer n.In particular,a left R-module M is FP-injective if and only if the Mn(R)-module Mn×ndefined by M is P-injective for each positive integer n.

Acknowledgments

The author would like to thank the referees for their useful comments.This research is supported by the Natural Science Foundation of Zhejiang Province,China(Grant No.LY18A010018).