$I-n$-Coherent Rings, $I-n$-Semihereditary Rings, and $I$-Regular Rings

Abstract

Let $R$ be a ring, let $I$ be an ideal of $R$, and let $n$ be a fixed positive integer. We define and study $I-n$-injective modules and $I-n$-flat modules. Moreover, we define and study left $I-n$-coherent rings, left $I-n$-semihereditary rings, and $I$-regular rings. By using the concepts of $I-n$-injectivity and $I-n$-flatness of modules, we also present some characterizations of the left $I-n$-coherent rings, left $I-n$-semihereditary rings, and $I$-regular rings.