Piecewise prime (PWP) module MR
is defined in terms of a set of triangulating idempotents in R. The class of PWP modules properly contains the class of prime modules. Some properties of these modules are investigated here.

1. Introduction

All rings are associative, and R denotes a ring with unity 1. The word ideal without the adjective right or left means two-sided ideal. The right annihilator of ideals of R is denoted by r.annR(I). A ring R is quasi-Baer (Baer) if the right annihilator of every right ideal (nonempty subset) of R is generated as a right ideal by an idempotent. We now recall a few definitions and results from [1] which motivated our study and serve as the background material for the present work. An idempotent e∈R is a left semicentral idempotent if exe=xe, for all x∈R. Similarly right semicentral idempotent can be defined. The set of all left (right) semicentral idempotents of R is denoted by Sl(R)(Sr(R)). An idempotent e∈R is semicentral reduced if Sl(eRe)={0,e}. If 1 is semicentral reduced, then R is called semicentral reduced. An ordered set {e1,…,en} of nonzero distinct idempotents of R is called a set of left triangulating idempotents of R if all the following hold:

e1+⋯+en=1,

e1∈Sl(R),

ek+1∈Sl(ckRck), where ck=1-(e1+⋯+ek) for 1≤k≤n.

From part (iii) of the previous definition, it can be seen that a set of left triangulating idempotents is a set of pairwise orthogonal idempotents. A set E={e1,…,en} of left triangulating idempotents of R is complete, if each ei is semicentral reduced. A (complete) set of right triangulating idempotents is defined similarly. The cardinalities of complete sets of left triangulating idempotents of R are the same and are denoted by τdim(R) [1, Theorem 2.10]. A ring R is called piecewise prime if there exists a complete set of left triangulating idempotents E={e1,…,en} of R, such that xRy=0 implies x=0 or y=0 where x∈eiRej and y∈ejRek for 1≤i,j,k≤n. In view of this definition we say a proper ideal I in R is a PWP ideal if there is a complete set of left triangulating idempotents E={e1,…,en}, such that xRy⊆I implies x∈I or y∈I, where x∈eiRej and y∈ejRek for 1≤i,j,k≤n. If R is PWP, then it is PWP with respect to any complete set of left triangulating idempotents of R; furthermore for a ring R with finite τdim(R), R is PWP if and only if R is quasi-Baer [1, Theorem 4.11].

A nonzero right R-module M is called a prime module if for any nonzero submodule N of M, r.annR(N)=r.annR(M), and a proper submodule P of M is a prime submodule of M if the quotient module M/P is a prime module. The notion of prime submodule was first introduced in [2, 3]; see also [4, 5]. It is easy to see that M is a prime R-module if and only if for any m∈M, and b∈R if mRb=0, then m=0 or Mb=0.

In this work the concept of prime modules is developed to piecewise prime modules as it is done for rings in [1]. Throughout this work it is considered that τdim(R) is finite.

2. Main ResultsDefinition 1.

Let M be an R-module and S=EndR(M).

M is a piecewise prime (PWP) R-module with respect to a complete set of left triangulating idempotents E={e1,…,en} of R, if for any m∈M, ei∈E, and b∈R,
(1)meiReib=0⇒mei=0orMeib=0.

Let N be a submodule of M. Then N is a piecewise prime submodule of M with respect to E if M/N is a PWP module with respect to E.

M is piecewise endoprime (PWEP) with respect to a complete set of left triangulating idempotents F={b1,…,bm} of S, such that for each nonzero submodule N⊆M, f∈S, and bi∈F, if fbiN=0, then fbi=0.

By Definition 1, N is a piecewise prime submodule of M with respect to a set of left triangulating idempotents E if for any m∈M, ei∈E, and b∈R,
(2)meiReib⊆N⇒mei∈NorMeib⊆N.

Example 2.

Let E={e1,…,en} be a complete set of left triangulating idempotents o R.

Let k1 and k2 be two fields and R=k1×k2. Then M=R/k1⊕R/k2 is not a prime module, but it is piecewise prime with respect to {(1,0),(0,1)}.

If M is a prime R-module, then it is piecewise prime with respect to any set of left triangulating idempotents of R.

Homomorphic image of MR needs to be PWP with respect to E. For example, ℤℤ is a PWP module with respect to {0,1}, but ℤ4 is not PWP because r.annr(2¯)≠r.annr(ℤ4).

Corollary 3.

If M is a PWP R-module with respect to E, then any submodule of M is PWP with respect to E.

Proof.

It can be seen by Definition 1.

Proposition 4.

Let R be a ring with finite triangulating dimension.

I is a PWP ideal of R if and only if R/I is a PWP R-module.

R is a PWP ring if and only if RR is PWP.

Proof.

The part one is obtained by Definition 1, and for second let I=0 in part one.

Proposition 5.

Let M be an R-module, and let E={e1,…,en} be a set of left triangulating idempotents of R. Then the following statements are equivalent:

M is PWP with respect to E;

for each N⊆M, ideal I in R, and ei∈E if NeiI=0 then Nei=0 or MeiI=0;

for each (m)⊆M, ideal (a) in R, and ei∈E if (m)ei(a)=0 then (m)ei=0 or Mei(a)=0.

Proof.

(1)⇒(2) If Nei≠0, then there exists n∈N, such that nei≠0, and for any b∈I, neiReib=0. By Definition 1, for each b∈I, Meib=0. This implies that MeiI=0.

(2)⇒(3) In (2), let N=(m) and I=(a).

(3)⇒(1) Let meiReib=0 where m∈M, ei∈E, and b∈R. Thus meiReiReibR=0 or (mei)ei(eib)=0. By (3), (mei)ei=0 or Mei(eib)=0. This implies that mei=0 or Meib=0.

Proposition 6.

Let M be an R-module, S=EndR(M), let E={e1,…,en} be a complete set of left triangulating idempotents of R, and let F={b1,…,bm} be a complete set of left triangulating idempotents of S.

M is a PWP R-module with respect to E if and only if for each N⊆M with Nei≠0, annr(Nei)=annr(Mei).

If MR is PWP R-module with respect to E, then annr(M) is a PWP ideal of R with respect to E.

If MR is PWEP with respect to F and retractable, then annr(M) is a PWP ideal of R with respect to E.

Proof.

(1) If b∈annr(Nei), then there exists n∈N, such that neiRei≠0 and neiReib=0. Since M is PWP R-module with respect to E by Definition 1, Meib=0. Hence annr(Nei)=annr(Mei). Conversely let meiReib=0 where ei∈E, m∈M, b∈R, and mei≠0. Thus b∈annr((meiR)ei) which means b∈annr(Mei) or Meib=0.

(2) Let IeiJ⊆annr(M) and Iei⊈annr(M). Since (MIei)eiJ=0, and M is a PWP R-module with respect to E, by Proposition 5, MeiJ=0. Thus eiJ⊆annr(M). This implies that annr(M) is a PWP ideal of R with respect to E.

(3) Let IeiJ⊆annr(M) where Iei,eiJ⊈annr(M). Since M is retractable, then there exists a nonzero homomorphism f:M→MIei. There exists bj∈F, such that fbj≠0. Since IeiJ⊆annr(M), fbjMeiJ=0. By assumption M is PWEP with respect to F. This implies that fbj=0 which is a contradiction. Hence annr(M) is a PWP ideal of R with respect to E.

A module MR is called retractable if for any nonzero submodule N of M, HomR(M,N)≠0.

Theorem 7.

Let M be an R-module, S=EndR(M), and let F={b1,…,bm} be a complete set of left triangulating idempotents of S.

If MS is a PWP module with respect to F, then S is a PWP ring. The converse is true when MR is retractable.

MS is a PWP module with respect to F, if and only if MR is PWEP with respect to F.

Proof.

(1) Let fbiSbig=0 where f,g∈S, bi∈F and big≠0. Thus there exists m∈M, such that bigm≠0 and fbiSbigm=0. Since MS is PWP with respect to F, fbiM=0 which means fbi=0. Conversely let fbiSbim=0 and bim≠0. Since MR is retractable, there exists a nonzero homomorphism big∈HomR(M,bimR). Thus fbiSbig=0. Since S is PWP, fbi=0.

(2) Assume M is a PWPS-module with respect to F. Let N⊆M and fbiN=0 where f∈S and bi∈F. Since MS is PWP, by Proposition 6(1), fbiM=0. Thus fbi=0. Conversely assume MR be PWEP with respect to F. Let fbiSbim=0 where f∈S, bi∈F, m∈M, and bim≠0. If N=Sbim, then fbiN=0. This implies that fbi=0 or fbiM=0. Hence MS is PWP with respect to F.

Let M be a right R-module with S=EndR(M). Then MR is called a quasi-Baer module, if for any N⊆SM, l.annS(N)=Se, where e=e2∈S [6].

Corollary 8.

Let M be a retractable R-module, S=EndR(M), and let F={b1,…,bm} be a complete set of left triangulating idempotents of S. Then the following statements are equivalent:

MR is a PWEP module with respect to F;

MS is a PWP module with respect to F;

MR is quasi-Baer.

Proof.

(1)⇔(2) This is evident by Theorem 7(2).

(2)⇔(3) By [6, Proposition 4.7], MR is quasi-Baer if and only if S is quasi-Baer. By [1, Theorem 4.11], S is PWP with respect to F if and only if S is quasi-Baer. The result is obtained by Theorem 7(1).

Proposition 9.

Let Λ be an index set, and let E={e1,…,en} be a complete set of left triangulating idempotents of R.

Let M=⊕λ∈ΛMλ. M is PWP with respect to E if and only if for each λ∈Λ, Mλ is PWP with respect to E.

Let M=∏λ∈ΛMλ. M is PWP with respect to E if and only if for each λ∈Λ, Mλ is PWP with respect to E.

Proof.

(1) Assume M is PWP with respect to E. If mλeiReib=0, where mλ∈Mλ, ei∈E, and b∈R then (0,…,mλ,0,…,0)eiReib=0. Since M is PWP, (0,…,mλ,0,…,0)ei=0 or Meib=0. This implies that mλei=0 or Mλeib=0 which means for each λ∈Λ, Mλ is PWP with respect to E. Conversely assume that for each λ∈Λ, Mλ is PWP with respect to E, and (m1,…,mn,0,…)eiReib=0. This implies that mλeiReib=0. Since Mλ is PWP with respect to E, mλei=0 or Meib=0. Hence (m1,…,mn,0,…)ei=0 or Meib=0. Thus M is PWP with respect to E.

(2) It can be seen by similar method as in part (1).

Corollary 10.

Let E={e1,…,en} be a complete set of left triangulating idempotents of R, let M be an R-module, and let F be a free R-module.

R is quasi-Baer if and only if F is a PWP module with respect to E.

M is PWP with respect to E if and only if F⊗R M is PWP with respect to E.

Proof.

It follows by [1, Theorem 4.11] and Proposition 9.

Proposition 11.

Let M be an R-module, and S=EndR(M). Then MS is prime if and only if τdim(S)=1, and MR is quasi-Baer.

Proof.

(⇒) Since M is a prime S-module, then for each N⊆M, l.annS(N)=l.annS(M)=0. This implies that MR is quasi-Baer. If e2=e∈S, then M=eM⊕(1-e)M. Since MS is prime, l.annS(eM)=l.annS((1-e)M)=l.annS(M). This implies that e=1 or e=0. Thus τdim(S)=1.

(⇐) Let N be any submodule of MS. Since MR is quasi-Baer, l.annS(N)=Se, where e∈Sr(S). Since τdim(S)=1, e∈{0,1}. If e=1, then N=0. Thus e=0. This implies that for each nonzero submodule N⊆SM, l.annS(N)=l.annS(M)=0. This means MS is prime.

It is folklore that prime radical plays an important role in the study of rings [7]. Following this concept is developed for modules of course by using a complete set of left triangulating idempotents of R.

Definition 12.

Let M be an R-module, let N be a proper submodule of M, and let E={e1,…,en} be a complete set of left triangulating idempotents of R.

The piecewise prime radical of N in M with respect to E is denoted by PRad(N) and is defined to be the intersection of all piecewise prime submodules of M with respect to E containing N.

PRad(M) means the intersection of all piecewise prime submodules of M with respect to E. If M has no piecewise prime submodule with respect to E, then PRad(M)=M.

Proposition 13.

Let N be a submodule of R-module M.

If N is a submodule of R-module M, then PRad(N)⊆PRad(M).

If PRad(M)=K, then PRad(M/K)=0.

If M=⊕i∈IMi is a direct sum of submodules Mi, then(3)PRad(M)=⨁i∈IPRad(Mi).

Proof.

Let E={e1,…,en} be a complete set of left triangulating idempotents of R.

Let K be any piecewise prime submodule of M with respect to E. If N⊆K, then PRad(N)⊆K. If N⊈K, then by the definition it is easy to see that N∩K is a piecewise prime submodule of N with respect to E. Thus PRad(N)⊆(K∩N)⊆K. Hence PRad(N)⊆PRad(M).

Let P/K be a piecewise prime submodules of M/K with respect to E. By definition (M/K)/(P/K) is a piecewise prime module with respect to E. Thus M/P is a a piecewise prime module with respect to E. This implies that P is a piecewise prime submodules of M with respect to E. Hence PRad(M/K)=0.

By (1) for each i∈I, PRad(Mi)⊆PRad(M). This implies that
(4)⨁i∈IPRad(Mi)⊆PRad(M).

Let (mi)i∈I∈M∖⊕i∈IPRad(Mi). Then there exists i∈I, such that mi∉PRad(Mi). By the definition there exists a piecewise prime submodule Ni⊆Mi with respect to E, such that mi∉Ni. If K=Ni⊕(⊕i≠jMj), then K is a piecewise prime submodule of M with respect to E, and m∉K. Thus m∉PRad(M). It means that
(5)PRad(M)=⨁i∈IPRad(Mi).BirkenmeierG. F.HeatherlyH. E.KimJ. Y.ParkJ. K.Triangular matrix representationsDaunsJ.Prime modulesFellerE. H.SwokowskiE. W.Prime modulesBehboodiM.KaramzadehO. A. S.KoohyH.Modules whose certain submodules are primeBehboodiM.KoohyH.On minimal prime submodulesRizviS. T.RomanC. S.Baer and quasi-Baer modulesLamT. Y.