Semi r-ideals of commutative rings

TL;DR

This paper introduces semi r-ideals in commutative rings, generalizing r-ideals and semiprime ideals, and explores their properties, characterizations, and extensions to modules across various ring constructions.

Contribution

It defines semi r-ideals, studies their properties, and extends the concept to semi r-submodules, providing a comprehensive framework for this new class of ideals.

Findings

Semi r-ideals generalize r-ideals and semiprime ideals.

Characterizations of semi r-ideals under various ring constructions.

Extension of semi r-ideals to semi r-submodules of modules.

Abstract

For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if whenever $a^{2}\in I$ and $Ann_{R}(a)=0$, then $a\in I$. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi $r$-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi $r$-ideals of rings to semi $r$-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the $D$-annihilator condition and justify when they are semi $r$-submodules.