Covers and Envelopes by Submodules or Quotient-Modules

Abstract

Let $R$ be a ring, $\mathcal{X}$ a class of left $R$ -modules, $\mathcal{S}$ the class of submodules of $\mathcal{X}$ , and $\mathcal{Q}$ the class of quotient-modules of $\mathcal{X}$ . It is shown that $\mathcal{S}$ $\left(\mathcal{Q}\right)$ is precovering (preenveloping) if and only if every injective (projective) left $R$ -module has an $\mathcal{X}$ -precover ( $\mathcal{X}$ -preenvelope). Both epic and monic $\mathcal{S}$ -(pre) covers ( $\mathcal{Q}$ -(pre) envelopes) are studied. Moreover, some applications are given. In particular, it is proven that the injective envelope of any projective left $R$ -module is projective if and only if the class of quotient-modules of projective and injective left $R$ -modules is monic preenveloping.