Weakly J-submodules of Modules over Commutative Rings

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. By $J(R),$ we denote the Jacobson radical of $R$. The purpose of this paper is to introduce the concept of weakly $J$-submodules generalizing $J$-submodules. We call a proper submodule $N$ of $M$ a weakly $J$-submodule if whenever $0\neq rm\in N$for$r\in R$and$m\in M$,then$r\in(J®M:M)$or$m\in N.$ Various properties and characterizations of weakly $J$-submodules are investigated especially in the case of multiplication modules.