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# Coordinatization Theorems For Graded Algebras

Published:2002-12-01
Printed: Dec 2002
• Bruce Allison
• Oleg Smirnov
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## Abstract

In this paper we study simple associative algebras with finite $\mathbb{Z}$-gradings. This is done using a simple algebra $F_g$ that has been constructed in Morita theory from a bilinear form $g\colon U\times V\to A$ over a simple algebra $A$. We show that finite $\mathbb{Z}$-gradings on $F_g$ are in one to one correspondence with certain decompositions of the pair $(U,V)$. We also show that any simple algebra $R$ with finite $\mathbb{Z}$-grading is graded isomorphic to $F_g$ for some bilinear from $g\colon U\times V \to A$, where the grading on $F_g$ is determined by a decomposition of $(U,V)$ and the coordinate algebra $A$ is chosen as a simple ideal of the zero component $R_0$ of $R$. In order to prove these results we first prove similar results for simple algebras with Peirce gradings.
 MSC Classifications: 16W50 - Graded rings and modules