JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY, vol.54, no.4, pp.1281-1299, 2017 (Journal Indexed in SCI)
A submodule N of a module M is called 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 S - Closed of short exact sequences determined by S-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class < S - Closed > containing S - Closed in terms of 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 torsionsplitting. In the second part, coprojective modules of this class which we call 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 < S - Closed > coincides with the class of pure-exact sequences of modules if and only if R is a two-sided hereditary, twosided CS-ring and every singular right module is a direct sum of finitely presented modules.