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Gabriel and Krull dimensions of modules over rings graded by finite groups


Authors: Piotr Grzeszczuk and Edmund R. Puczyłowski
Journal: Proc. Amer. Math. Soc. 105 (1989), 17-24
MSC: Primary 16A55; Secondary 06C05
DOI: https://doi.org/10.1090/S0002-9939-1989-0973835-X
MathSciNet review: 973835
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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 $ {\text{G}}{\dim _R}M = {\text{G}}{\dim _{{R_e}}}M,{\text{K}}{\dim _R}M = {\text{K}}{\dim _{{R_e}}}M$ and $ {\text{N}}{\dim _R}M = {\text{N}}{\dim _{{R_e}}}M$, where $ {\text{G}}\dim ,{\text{K}}\dim $ and $ {\text{N}}\dim $ 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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DOI: https://doi.org/10.1090/S0002-9939-1989-0973835-X
Article copyright: © Copyright 1989 American Mathematical Society

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