The Generalization of Prime Modules

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.