Vol. 39, No. 1, 1971

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On splitting in hereditary torsion theories

Robert L. Bernhardt, III

Vol. 39 (1971), No. 1, 31–38

Let (𝒯 ,) denote a hereditary torsion theory for the category of modules over a ring R. In this paper the splitting of projective modules is studied, and it is shown that this is not equivalent to the splitting of quasi-projective modules. In addition, situations arising from the class of torsion modules 𝒯 (or the class of torsionfree modules ) being contained in the injective or in the projective modules are considered, and several conditions sufficient for an especially strong form of splitting are given. Finally when 𝒯 is closed under injective envelopes the following is shown: every module splits if R is an artinian generalized uniserial ring, and projective modules split if R is a QF 2 ring.

Mathematical Subject Classification
Primary: 16A50
Received: 13 October 1970
Revised: 29 April 1971
Published: 1 October 1971
Robert L. Bernhardt, III