Generalizations of r-ideals of commutative rings

TL;DR

This paper introduces a generalized concept of r-ideals in commutative rings, explores their properties, and applies these ideas to trivial ring extensions and the characterization of total quotient rings.

Contribution

It extends the notion of r-ideals to a broader class called -r-ideals, analyzing their properties and applications in ring extensions and quotient rings.

Findings

Many properties of -r-ideals are established

Characterizations of total quotient rings using -r-ideals are provided

Applications in trivial ring extensions are discussed

Abstract

In this study, we present the generalization of the concept of $r$-ideals in commutative rings with nonzero identity. Let $R$ be a commutative ring with $0\neq1$ and $L(R)$ be the lattice of all ideals of $R$. Suppose that $\phi:L(R)\rightarrow L(R)\cup\left\{\emptyset\right\}$ is a function. A proper ideal $I$ of $R$ is called a $\phi-r$-ideal of $R$ if whenever $ab\in I$ and $Ann(a)=(0)$ imply that $b\in I$ for each $a,b\in R.$ In addition to giving many properties of $\phi-r$-ideal, we also examine the concept of $\phi-r$-ideal in trivial ring extension and use them to characterize total quotient rings.