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

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.