  Let $R$ be a commutative ring with non-zero identity. In this paper, we
introduce the concept of weakly $J$-ideals as a new generalization of
$J$-ideals. We call a proper ideal $I$ of a ring $R$ a weakly $J$-ideal if
whenever $a,b\in R$ with $0\neq ab\in I$ and $a\notin J(R)$, then $a\in I$.
Many of the basic properties and characterizations of this concept are studied.
We investigate weakly $J$-ideals under various contexts of constructions such
as direct products, localizations, homomorphic images. Moreover, a number of
examples and results on weakly $J$-ideals are discussed. Finally, the third
section is devoted to the characterizations of these constructions in an
amagamated ring along an ideal.
