Modules (co)reduced relative to another module

Abstract

Let $R$ be a commutative unital ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a fixed $R$-module. We introduce and study generalisations of $\mathfrak{a}$-reduced modules, ${\mathfrak{R}}_{\mathfrak{a}}$ and $\mathfrak{a}$-coreduced modules, ${\mathfrak{C}}_{\mathfrak{a}}$ studied in the literature. They are called modules $\mathfrak{a}$-reduced with respect to $M$, ${\mathfrak{R}}_{\mathfrak{a}}^M$ and modules $\mathfrak{a}$-coreduced with respect to $M$, ${\mathfrak{C}}_{\mathfrak{a}}^M$. The paper lifts several known results about ${\mathfrak{R}}_{\mathfrak{a}}$ and ${\mathfrak{C}}_{\mathfrak{a}}$ to larger classes of modules ${\mathfrak{R}}_{\mathfrak{a}}^M$ and ${\mathfrak{C}}_{\mathfrak{a}}^M$ respectively. Results about ${\mathfrak{R}}_{\mathfrak{a}}$ and ${\mathfrak{C}}_{\mathfrak{a}}$ can be retrieved by taking $M=R$.

19 pages