The singular submodule splits off

Abstract

In the usual torsion theory over a commutative integral domain R, a significant place is occupied by the question of when “the torsion submodule t(M) of an R-module M is a direct summand of M” (hereafter referred to as “the condition”). Kaplansky [IO] h as shown that if the condition is satisfied by every finitely generated R-module, then R is a Pri.ifer domain; the converse is well known [2], Chapt. VII. Prop. 11.1. If the condition is satisfied by every R-module M, whose torsion submodule is of bounded order, Chase [3], Theorem 4.3, has shown that R is, then, a Dedekind domain; the converse has been shown by Kaplansky [II], Theorem 5. Finally, if the condition is satisfied by every R-module, Rotman [16] has shown that R is, then, a field. A well-known extension of the notion of the torsion submodule for modules over arbitrary rings is the notion of the singular submodule of a module [5, 7,9]. It is the purpose of this paper to investigate “the condition” of the preceding paragraph, in the context of the singular theory over a nonsingular commutative ring R. In short, if M is an R-module and Z(M) its singular submodule, we study the condition: “Z(M) is a direct summand of M”. It is worthwhile to note that the commutative nonsingular rings are precisely the commutative semi-prime rings [12], Ex. 1, p. 108. Throughout this paper, unless otherwise indicated, a ring R is a commutative ring with identity; all modules are unitary. For all homological notions, used in this paper, the reader is referred to [2].