Weakly J-ideals of Commutative Rings

TL;DR

This paper introduces weakly J-ideals as a new generalization of J-ideals in commutative rings, exploring their properties, characterizations, and behavior under various ring constructions.

Contribution

It defines weakly J-ideals, studies their fundamental properties, and examines their behavior in different ring constructions, extending the theory of J-ideals.

Findings

Weakly J-ideals generalize J-ideals in commutative rings.

Characterizations of weakly J-ideals are established.

Behavior under direct products, localizations, and homomorphic images is analyzed.

Abstract

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.