On Jacobson – Small Submodules

Abstract

Let R be an associative ring with identity and let M be a unitary left R–module. As a generalization of small submodule , we introduce Jacobson–small submodule (briefly J–small submodule ) . We state the main properties of J–small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobson–hollow modules ( briefly J–hollow ) . We give a characterization of J–hollow modules and gives conditions under which the direct sum of J–hollow modules is J–hollow . We define J–supplemented modules and some types of modules that are related to J–supplemented modules and introduce properties of this types of modules . Also we discuss the relation between them with examples and remarks are needed in our work.

F-J-semi Regular Modules

“…In order to introduce semi regular module , we use the concept of -small submodule that is appeared in [3]. Definition 2.1 [3] : Let be any -module a submodule of is called Jacobson small (for short -small , denoted by ) if whenever , , and ( ) , then . Lemma 2.2 [3] : Let , be two submodules of an -module , if and ( ) , then ( ) .…”

“…The sum of all small submodules is called the jacobson radical of which is denoted by ( ) [ 2 ] . In [3], authors introduced -small submodule. A submodule of is called -small if whenever with ( ) implies that .…”

Jacobson Hopfian modules

“…The radical of an R-module M deőned as a dual of the socle of M , is the intersection of all maximal submodules of M , taking Rad(M ) = M when M has no maximal submodules. A submodule K of M is said to be Jacobson-small in M (K ≪ J M ), in case M = K + L with Rad(M/L) = M/L, implies M = L (see [10]). It is clear that if A is a small submodule of M , then A is a Jacobson-small submodule of M , but the converse is not true in general.…”

“…It is clear that if A is a small submodule of M , then A is a Jacobson-small submodule of M , but the converse is not true in general. By [10], if Rad(M ) = M and K ⩽ M , then K is small in M if and only if K is Jacobson-small in M .…”

D_j -Supplemented Modules

“…(𝒲) . 𝒜 + 𝐶 ↪ 𝒲 1 + 𝐶 and ℬՈ(𝒜 + 𝒞) ≪ 𝑗 ℬՈ(𝒲 1 + 𝒞) ≪ 𝑗 𝐷 𝑗 (ℬ) , ℬՈ(𝒜 + 𝒞) ≪ 𝑗 𝐷 𝑗 (ℬ) by [5]. (ℬ + 𝒜)Ո𝒞 ≪ 𝑗 ℬ Ո(𝒜 + 𝒞) + 𝒜Ո(ℬ + 𝒞) ≪ 𝑗 𝐷 𝑗 (ℬ) + 𝐷 𝑗 (𝒜) so 𝒜Ո(ℬ + 𝒞) ≪ 𝑗 𝐷 𝑗 (ℬ + 𝒜) by [1] Lemma 2.9: if 𝒲 any module and ℬ is a 𝐷 𝑗 -'supplement of 𝒜 in 𝒲 for 𝒩 ↪ ℬ , 𝒩 ≪ 𝑗 𝒲 if and only if 𝒩 ≪ 𝑗 ℬ . )…”