The concept of a torsion theory
(𝒯 ,ℱ) for left R-modules has been defined by S. E. Dickson. A torsion theory is
called splitting if it has the property that the torsion submodule of every
left R-module is a direct summand. Under restrictive hypotheses on the
ring R, several specific splitting theories have previously been examined.
This paper continues the investigation to more general classes of torsion
theories. In the first section, comparisons are made between injective modules
and torsion modules for a splitting theory, and the following results are
obtained: (1) A torsion class 𝒯 is closed under taking injective envelopes
if and only if the maximal 𝒯 -torsion submodule of an injective module is
injective. (2) If (𝒯 ,ℱ) is splitting and R ∈ℱ then inj dim(T) ≦ 1 for all
T ∈𝒯 . (3) If (𝒯 ,ℱ) is splitting and hereditary and if R ∈ℱ, then every
homomorphic image of a 𝒯 -torsion injective module is injective. In §2 it is
shown that rings R, for which R has zero singular ideal and Goldie’s torsion
theory is splitting, have the property: 1. gl. dimR ≦ 2. It is shown that
the relative homological dimension arising from a hereditary torsion theory
often gives information about splitting, especially when this dimension is
zero. In the final sections, the zero-dimensionality of a hereditary torsion
theory is discussed and related to results of J. P. Jans. The rings, all of whose
hereditary torsion theories have dimension zero, are characterized as direct sums
of finitely many right perfect rings, each of which has a unique maximal
ideal.
