Generalizations of r-ideals of commutative rings

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 $0\ne 1$ and $L(R)$ be the lattice of all ideals of $R$. Suppose that $\varphi :L(R)\to L(R)\cup \left\{\varnothing \right\}$ is a function. A proper ideal $I$ of $R$ is called a $\varphi -r$-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 $\varphi -r$-ideal, we also examine the concept of $\varphi -r$-ideal in trivial ring extension and use them to characterize total quotient rings.