Torsion theories and semihereditary rings

Abstract

Introduction. Let R be an associative ring with identity and let RAN (respectively 9ifR) denote the category of unitary left (respectively right) R-modules. Dickson [4] has given an axiomatic treatment of torsion for abelian categories. Specializing his definition, a torsion theory for RA is defined to be a pair (3, 5) of classes of left R-modules such that (a) 3n5CM=O0}, (b) 3 is closed under homomorphic images, (c) 5f is closed under submodules, (d) for every left R-module M there exists a submodule T(M) of M with T(M) Ez and M/T(M) E . Throughout this paper two additional properties are required of a torsion theory (e) RREf, (f) 5 is closed under submodules. A torsion theory for OJMR is similarly defined. If (r, 5) is a torsion theory those modules in 3 are said to be torsion and those in 5f are said to be torsion-free. Specific examples of torsion theories having these properties are the usual torsion theory for abelian groups, the torsion theory for left R-modules over a left Ore ring studied by Levy [9] and the E(R)-torsion theory considered by Jans [7].