ON S-CLOSED SUBMODULES
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A submodule $N$ of a module $M$ is called $\mathcal{S}$-closed (in $M$) if $M/N$ is nonsingular. It is well-known that the class $\closed$ of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-\closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class $\langle\mathcal{S}-\closed\rangle$ containing $\mathcal{S}-\closed$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring $R$, it coincides with the proper class generated by neat submodules if and only if $R$ is a right SI-ring. In abelian groups, the elements of this class are exactly torsion-splitting. In the second part, coprojective modules of this class which we call \emph{ec-flat} modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring $R$ of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class $\langle\mathcal{S}-\closed\rangle $ coincides with the class of pure-exact sequences of modules if and only if $R$ is a two-sided hereditary, two-sided $CS$-ring and every singular right module is a direct sum of finitely presented modules.
Abstract
A submodule $N$ of a module $M$ is called $\mathcal{S}$-closed (in $M$) if $M/N$ is nonsingular. It is well-known that the class $\closed$ of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-\closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class $\langle\mathcal{S}-\closed\rangle$ containing $\mathcal{S}-\closed$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring $R$, it coincides with the proper class generated by neat submodules if and only if $R$ is a right SI-ring. In abelian groups, the elements of this class are exactly torsion-splitting. In the second part, coprojective modules of this class which we call \emph{ec-flat} modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring $R$ of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class $\langle\mathcal{S}-\closed\rangle $ coincides with the class of pure-exact sequences of modules if and only if $R$ is a two-sided hereditary, two-sided $CS$-ring and every singular right module is a direct sum of finitely presented modules.
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