$\delta$-$J$-ideals of commutative rings

TL;DR

This paper introduces the concept of $oldsymbol{ extdelta}$-$oldsymbol{J}$-ideals in commutative rings, exploring their properties and behavior under various ring-theoretic operations.

Contribution

It defines $oldsymbol{ extdelta}$-$oldsymbol{J}$-ideals and investigates their properties, including localization, homomorphic images, and idealization, expanding the theory of ring ideals.

Findings

$oldsymbol{ extdelta}$-$oldsymbol{J}$-ideals are characterized and their properties are established.

The behavior of $oldsymbol{ extdelta}$-$oldsymbol{J}$-ideals under localization and homomorphisms is analyzed.

The paper provides foundational results for further study of these ideals in commutative algebra.

Abstract

Let $\mathcal{I}(R)$ be the set of all ideals of a ring $R$, $\delta$ be an expansion function of $\mathcal{I}(R)$. In this paper, the $\delta$-$J$-ideal of a commutative ring is defined, that is, if $a, b\in R$ and $ab\in I\in \mathcal{I}(R)$, then $a\in J(R)$ (the Jacobson radical of $R$) or $b\in \delta(I)$. Moreover, some properties of $\delta$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.