  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 $r\in(J(R)M:M)$ or $m\in N.$ Various
properties and characterizations of weakly $J$-submodules are investigated
especially in the case of multiplication modules.
