The torsion submodule splits off

Abstract

A classical question for modules over an integral domain is, "When is the torsion submodule t(A) of a module A a direct summand of AT' A module is said to split when its torsion submodule is a direct summand. Kaplansky has shown [-14] that if R is a Dedekind domain, then every module whose torsion submodule is of bounded order splits. The converse of this result has been shown by Chase [5]. Results of [15] and [3] show that every finitely generated module splits if and only if R is a Priifer domain. Finally, if every R-module splits, then Rotman has shown [19] that R must be a field. Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Almost all of these are special cases of "torsion theories" in the sense of Dickson [6]. Moreover, most of these torsion theories are hereditary (i.e., the submodule of a torsion module is torsion); and hereditary torsion classes are classes of negligible modules associated with a topologizing and idempotent filter of left ideals in the sense of Gabriel [12] (also see [17]). Some recent papers ([4], [7], and [11]) have dealt with splitting results for specific hereditary torsion theories over certain commutative rings. The main purpose of this paper is to continue the investigation of the splitting properties of a torsion theory of modules over a commutative ring. Some characterizations for the splitting of modules, whose torsion submodules have bounded order, are obtained (see definition of bounded order below and Theorems 2.2 and 4.6). In particular, these results generalize the abovementioned theorems of Chase [5] and Kaplansky [14]. Our results show that the splitting of modules whose torsion submodules have bounded order frequently forces non-zero torsion modules to have non-zero socles. This increases our interest in the smallest hereditary torsion class 6e containing the simple modules. The class ow has previously been used in the study of commutative Noetherian rings and (left) perfect rings (e.g., see [7] and [10] and their references). For a commutative ring R, we show that 6e(A) is a summand of each R-module A if and only if non-zero R-modules have non-zero socles. This generalizes the main results of [7] and [11]; moreover, in case R is a Dedekind domain, our result coincides with the above result of Rotman. We also examine the properties of a splitting hereditary torsion theory of modules over a local ring R (unique maximal left ideal). We show that if R possesses a non-trivial torsion theory (~--, #-) such that every finitely generated module splits, then R is an integral domain, and (Y-, ~ is Goldie's torsion