Very good gradings on matrix rings are epsilon-strong
LINEAR & MULTILINEAR ALGEBRA(2024)
Abstract
We investigate properties of group gradings on matrix rings $ M_n(R) $ Mn(R), where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on $ M_n(R) $ Mn(R) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that $ M_n(R) $ Mn(R) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when $ M_n(R) $ Mn(R) is an epsilon-crossed product. Our results are illustrated by several examples.
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Key words
Matrix ring,good grading,very good grading,epsilon-strongly graded ring,unital partial crossed product
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