Vol. 24, No. 2, 1968

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Goldie’s torsion theory and its derived functor

John Suemper Alin and Spencer Ernest Dickson

Vol. 24 (1968), No. 2, 195–203

In this paper the global dimension of any complete, wellpowered abelian category wilh injective envelopes in calculated relative to the torsion theory of A. W. Goldie and is found to be always one or zero. The rings R such that the left module category Rhas global dimension zero are precisely those such that every module having zero singular submodule is injective. These rings are characterized as being of the form T S (ring direct sum) where T is a ring having essential singular ideal and S is semi-simple with minimum condition. The rings with essential singular ideal are precisely those which are torsion as left modules over themselves.

Mathematical Subject Classification
Primary: 18.15
Received: 14 March 1967
Published: 1 February 1968
John Suemper Alin
Spencer Ernest Dickson