MMN-1566

For an integral domain $R$ we consider the closures $\widehat{M}$ $(\widehat{M}_{r}, r\in R)$ of a submodule $M$ of an $R$-module $N$ consisting of elements $n$ of $N$ with $tn\in M$ $(r^{m}n\in M)$ for some nonzero $t\in R$ $(m\in \mathbb{Z}^{+})$ and its connections with usual closure $\overline{M}$ of $M$ in $N$. Using these closures we study the closures $\hat{\mathcal{P}}$ and $\hat{\mathcal{P}}_{r}$ of a proper class $\mathcal{P}$ of short exact sequences and give a decomposition for the class of quasi-splitting short exact sequences of abelian groups into the direct sum of ``$p$-closures" of the class $\Split$ of splitting short exact sequences and description of closures of some classes. In the general case of an arbitrary ring we generalize these closures of a proper class $\mathcal{P}$ by means of homomorphism classes $\mathcal{F} $ and $\mathcal{G}$ and prove that under some conditions this closure $\hat{\mathcal{P}}_{\mathcal{F}}^{\mathcal{G}}$ is a proper classes.

Vol. 17 (2016), No. 2, pp. 723-738

DOI: 10.18514/MMN.2017.1566

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