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Gabriel and Krull Dimensions of Modules Over Rings Graded by Finite Groups

Piotr Grzeszczuk and Edmund R. Puczyłowski
Proceedings of the American Mathematical Society
Vol. 105, No. 1 (Jan., 1989), pp. 17-24
DOI: 10.2307/2046727
Stable URL: http://www.jstor.org/stable/2046727
Page Count: 8
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Gabriel and Krull Dimensions of Modules Over Rings Graded by Finite Groups
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Abstract

Let $R$ be a ring graded by a finite group $G$ with the identity component $R_e$ and let $M$ be a left $R$-module. It is proved that $\mathrm{Gdim}_R M = \mathrm{Gdim}_{R_e} M, \mathrm{Kdim}_R M = \mathrm{Kdim}_{R_e} M$ and $\mathrm{Ndim}_R M = \mathrm{Ndim}_{R_e} M$, where Gdim, Kdim and Ndim denote, respectively, Gabriel, Krull and dual Krull dimensions. The proofs are based on the use of lattice theory, a method which also gives alternative proofs of known results about normalizing extensions.

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