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The G-Function of Macrae

Published online by Cambridge University Press:  20 November 2018

David E. Rush
David E. Rush*
Affiliation:
University of California, Riverside, Riverside, California
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Let R be a commutative ring with identity. A finitely generated R-module M is called a torsion module if the annihilator of M contains a non zero-divisor. In [18] MacRae proved the following

Theorem. If R is a noetherian ring then there is a map G with the following properties from the class of torsion R-modules of finite homological dimension to the set of integral invertible ideals of R.

(i) If M is a finitely generated torsion R-module with homological dimension ≦ 1 then G(M) = F(M), the first Fitting ideal of M.

(ii) If S is a multiplicative subset of R then G(Ms) = G(M) s.

(iii) If 0 → L → M → N → 0 is an exact sequence of torsion modules of finite homological dimension then G(M) = G(L)G(N).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 4 , August 1974 , pp. 854 - 865
Copyright
Copyright © Canadian Mathematical Society 1974

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