#### Abstract

Piecewise prime (PWP) module is defined in terms of a set of triangulating idempotents in . 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 denotes a ring with unity . The word ideal without the adjective right or left means two-sided ideal. The right annihilator of ideals of is denoted by . A ring is - () if the right annihilator of every right ideal (nonempty subset) of is generated as a right ideal by an idempotent. We now recall a few definitions and results from  which motivated our study and serve as the background material for the present work. An idempotent is a left semicentral idempotent if , for all . Similarly right semicentral idempotent can be defined. The set of all left (right) semicentral idempotents of is denoted by . An idempotent is semicentral reduced if . If is semicentral reduced, then is called semicentral reduced. An ordered set of nonzero distinct idempotents of is called a set of left triangulating idempotents of if all the following hold:(i),(ii),(iii), where for .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 of left triangulating idempotents of is complete, if each is semicentral reduced. A (complete) set of right triangulating idempotents is defined similarly. The cardinalities of complete sets of left triangulating idempotents of are the same and are denoted by [1, Theorem 2.10]. A ring is called piecewise prime if there exists a complete set of left triangulating idempotents of , such that implies or where and for . In view of this definition we say a proper ideal in is a ideal if there is a complete set of left triangulating idempotents , such that implies or , where and for . If is , then it is with respect to any complete set of left triangulating idempotents of ; furthermore for a ring with finite , is if and only if is quasi-Baer [1, Theorem 4.11].

A nonzero right -module is called a prime module if for any nonzero submodule of , , and a proper submodule of is a prime submodule of if the quotient module 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 is a prime -module if and only if for any , and if , then or .

In this work the concept of prime modules is developed to piecewise prime modules as it is done for rings in . Throughout this work it is considered that is finite.

#### 2. Main Results

Definition 1. Let be an -module and . (1) is a piecewise prime () -module with respect to a complete set of left triangulating idempotents of , if for any , , and , (2)Let be a submodule of . Then is a piecewise prime submodule of with respect to if is a module with respect to . (3) is piecewise endoprime () with respect to a complete set of left triangulating idempotents of , such that for each nonzero submodule , , and , if , then .

By Definition 1, is a piecewise prime submodule of with respect to a set of left triangulating idempotents if for any , , and ,

Example 2. Let be a complete set of left triangulating idempotents o . (1)Let and be two fields and . Then is not a prime module, but it is piecewise prime with respect to . (2)If is a prime -module, then it is piecewise prime with respect to any set of left triangulating idempotents of . (3)Homomorphic image of needs to be with respect to . For example, is a module with respect to , but is not because .

Corollary 3. If is a PWP -module with respect to , then any submodule of is PWP with respect to .

Proof. It can be seen by Definition 1.

Proposition 4. Let be a ring with finite triangulating dimension. (1) is a PWP ideal of if and only if is a PWP -module.(2) is a PWP ring if and only if is PWP.

Proof. The part one is obtained by Definition 1, and for second let in part one.

Proposition 5. Let be an -module, and let be a set of left triangulating idempotents of . Then the following statements are equivalent:(1) is PWP with respect to ;(2)for each , ideal in , and if then or ;(3)for each , ideal in , and if then or .

Proof. If , then there exists , such that , and for any , . By Definition 1, for each , . This implies that .
In (2), let and .
Let where , , and . Thus or . By (3), or . This implies that or .

Proposition 6. Let be an -module, , let be a complete set of left triangulating idempotents of , and let be a complete set of left triangulating idempotents of . (1) is a PWP -module with respect to if and only if for each with , . (2)If is PWP -module with respect to , then is a PWP ideal of with respect to . (3)If is PWEP with respect to and retractable, then is a PWP ideal of with respect to .

Proof. (1) If , then there exists , such that and . Since is PWP -module with respect to by Definition 1, . Hence . Conversely let where , , , and . Thus which means or .
(2) Let and . Since , and is a PWP -module with respect to , by Proposition 5, . Thus . This implies that is a ideal of with respect to .
(3) Let where . Since is retractable, then there exists a nonzero homomorphism . There exists , such that . Since , . By assumption is with respect to . This implies that which is a contradiction. Hence is a ideal of with respect to .

A module is called retractable if for any nonzero submodule of , .

Theorem 7. Let be an -module, , and let be a complete set of left triangulating idempotents of . (1)If is a PWP module with respect to , then is a PWP ring. The converse is true when is retractable. (2) is a PWP module with respect to , if and only if is PWEP with respect to .

Proof. (1) Let where , and . Thus there exists , such that and . Since is with respect to , which means . Conversely let and . Since is retractable, there exists a nonzero homomorphism . Thus . Since is , .
(2) Assume is a   -module with respect to . Let and where and . Since is , by Proposition 6(1), . Thus . Conversely assume be with respect to . Let where , , , and . If , then . This implies that or . Hence is with respect to .

Let be a right -module with . Then is called a quasi-Baer module, if for any , , where .

Corollary 8. Let be a retractable -module, , and let be a complete set of left triangulating idempotents of . Then the following statements are equivalent:(1) is a PWEP module with respect to ;(2) is a PWP module with respect to ;(3) is quasi-Baer.

Proof. This is evident by Theorem 7(2).
By [6, Proposition 4.7], is quasi-Baer if and only if is quasi-Baer. By [1, Theorem 4.11], is with respect to if and only if is quasi-Baer. The result is obtained by Theorem 7(1).

Proposition 9. Let be an index set, and let be a complete set of left triangulating idempotents of . (1)Let . is PWP with respect to if and only if for each , is PWP with respect to . (2)Let . is PWP with respect to if and only if for each , is PWP with respect to .

Proof. (1) Assume is with respect to . If , where , , and then . Since is , or . This implies that or which means for each , is with respect to . Conversely assume that for each , is with respect to , and . This implies that . Since is with respect to , or . Hence or . Thus is with respect to .
(2) It can be seen by similar method as in part (1).

Corollary 10. Let be a complete set of left triangulating idempotents of , let be an -module, and let be a free -module. (1) is quasi-Baer if and only if is a PWP module with respect to . (2) is PWP with respect to if and only if     M is PWP with respect to .

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

Proposition 11. Let be an -module, and . Then is prime if and only if , and is quasi-Baer.

Proof. Since is a prime -module, then for each , . This implies that is quasi-Baer. If , then . Since is prime, . This implies that or . Thus .
Let be any submodule of . Since is quasi-Baer, , where . Since , . If , then . Thus . This implies that for each nonzero submodule , . This means is prime.

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

Definition 12. Let be an -module, let be a proper submodule of , and let be a complete set of left triangulating idempotents of . (1)The piecewise prime radical of in with respect to is denoted by and is defined to be the intersection of all piecewise prime submodules of with respect to containing . (2) means the intersection of all piecewise prime submodules of with respect to . If has no piecewise prime submodule with respect to , then .

Proposition 13. Let be a submodule of -module . (1)If is a submodule of -module , then .(2)If , then . (3)If is a direct sum of submodules , then

Proof. Let be a complete set of left triangulating idempotents of .(1)Let be any piecewise prime submodule of with respect to . If , then . If , then by the definition it is easy to see that is a piecewise prime submodule of with respect to . Thus . Hence .(2)Let be a piecewise prime submodules of with respect to . By definition is a piecewise prime module with respect to . Thus is a a piecewise prime module with respect to . This implies that is a piecewise prime submodules of with respect to . Hence .(3)By (1) for each , . This implies that Let . Then there exists , such that . By the definition there exists a piecewise prime submodule with respect to , such that . If , then is a piecewise prime submodule of with respect to , and . Thus . It means that