Vol. 34, No. 1, 1970

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Homological dimension and splitting torsion theories

Mark Lawrence Teply

Vol. 34 (1970), No. 1, 193–205

The concept of a torsion theory (𝒯 ,) 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 ,) is splitting and R ∈ℱ then inj dim(T) 1 for all T ∈𝒯 . (3) If (𝒯 ,) 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.

Mathematical Subject Classification
Primary: 16.90
Received: 20 February 1969
Published: 1 July 1970
Mark Lawrence Teply