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Algebra
Volume 2013 (2013), Article ID 581023, 4 pages
http://dx.doi.org/10.1155/2013/581023
Research Article

The Generalization of Prime Modules

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

Received 29 December 2012; Accepted 15 February 2013

Academic Editor: Masoud Hajarian

Copyright © 2013 M. Gurabi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1] 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 [1]. 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 [6].

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 [7]. 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

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