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 $M$, $\mathfrak{R}^{M}_{\mathfrak{ a}}$ and modules $\mathfrak{ a}$-coreduced with respect to $M$, $\mathfrak{C}^{M}_{\mathfrak{ a}}$. The paper lifts several known results about $\mathfrak{R}{\mathfrak{ a}}$ and $\mathfrak{C}{\mathfrak{ a}}$ to larger classes of modules $\mathfrak{R}^{M}_{\mathfrak{ a}}$ and $\mathfrak{C}^{M}_{\mathfrak{ a}}$ respectively. Results about $\mathfrak{R}{\mathfrak{ a}}$ and $\mathfrak{C}{\mathfrak{ a}}$ can be retrieved by taking $M= R$.