e*-Essential Small Submodules and e*-Hollow Modules

Abstract

The purpose of this paper is to introduce new concepts in a module  over a ring , the first one is called e*- small essential submodule, which is a generalization of the small submodule, the second concept is called e*-radical submodule which is a generalization of the radical submodule and the last concept is called e* - hollow module which is a generalization of the hollow module. We will prove some properties of all these concepts.

e*-Hollow-Lifting and Cofinitely e*-Lifting Modules

“…This section introduces generalizations for the e * -lifting module with specific properties. The characteristics of e * -essential small are listed below that appeared in [3].…”

“…4Z is not e * -essential small in Z. See [3]. (4) Every semisimple module is e * -hollow-lifting and every e * -hollow module is e * -hollow-lifting.…”

“…If S ∩ T = {0} for each cosingular T where 0 = T ≤ M , S is called an e * -essential submodule of M , symbolized by S ≤ e * M . In [3], Baanoon and Khalid used e *essential submodules to present a new class of submodules, a generalization of a small submodule, called e * -essential small. If S + T = M implies T = M for each T ≤ e * M , S is called an e * -essential small submodule of M symbolized by (S e * M ).…”

“…If S + T = M implies T = M for each T ≤ e * M , S is called an e * -essential small submodule of M symbolized by (S e * M ). The generalization of the radical submodule was introduced in [3], which is called e * -radical denoted by, Rad(M ) e * -essential maximal submodule of a module M . Equivalently, Rad(M )…”

“…If each proper submodule of M is e * -essential small, then M is anointed e * -hollow, where M is a nonzero module. See [3]. If module M has a direct summand T such that T ≤ S and S T M T for each submodules S, it is said to be lifting.…”