On Almost Semiprime Submodules

We introduce the concept of almost semiprime submodules of unitary modules over a commutative ring with nonzero identity. We investigate some basic properties of almost semiprime and weakly semiprime submodules and give some characterizations of them, especially for (finitely generated faithful) multiplication modules. 1. Introduction Throughout this paper, all rings are commutative rings with identity and all modules are unitary. Various generalizations of prime (primary) ideals are studied in [1–8]. The class of prime submodules of modules as a generalization of the class of prime ideals has been studied by many authors; see, for example, [9, 10]. Then many generalizations of prime submodules were studied such as weakly prime (primary) [11], almost prime (primary) [12], 2-absorbing [13], classical prime (primary) [14, 15], and semiprime submodules [16]. In this paper, we study weakly semiprime and almost semiprime submodules as the generalizations of semiprime submodules. Weakly semiprime submodules have been already studied in [17]. Here we first define the notion almost semiprime submodules and get a number of propensities of almost semiprime and weakly semiprime submodules. Also, we give some characterizations of such submodules in multiplication modules. Now we define the concepts that we will use. For any two submodules and of an -module , the residual of by is defined as the set which is clearly an ideal of . In particular, the ideal is called the annihilator of . Let be a submodule of and let be an ideal of ; the residual submodule of by is defined as . These two residual ideals and submodules were proved to be useful in studying many concepts of modules; see, for example, [18, 19]. A proper submodule of an -module is a prime submodule if, whenever for and , or . An -module is called a prime module if its zero submodule is a prime submodule. A proper submodule of an -module is called weakly prime (weakly primary) if , where and ; then or ( or ). A proper submodule of an -module is called almost prime (almost primary) if, whenever for and , or ( or ). A proper ideal of a commutative ring is called semiprime if , where and ; then . A proper submodule of an -module is called semiprime if, whenever , , and such that , . An -module is called a second module provided that, for every element , the -endomorphism of produced by multiplication by is either surjective or zero; this implies that is a prime ideal of and is said to be -second [20]. An -module is called a multiplication module provided that, for every submodule of , there exists an ideal of