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Coordinatization theorems for graded algebras. (English) Zbl 1018.16022

Let \(R\) be an algebra (not necessarily with unity) over a base ring \(\Phi\). A Peirce grading on \(R\) is a decomposition \(R=\bigoplus_{i,j=0}^nR_{i,j}\) as \(\Phi\)-modules, such that \(R_{i,j}R_{k,l}\subseteq\delta_{jk}R_{i,l}\) for any \(i,j,k,l\). This generalizes the Peirce decomposition associated to a complete set of orthogonal idempotents of an algebra with unity. The aim of the paper is to investigate \(\mathbb{Z}\)-gradings of simple algebras in connection with Peirce gradings. The authors present the construction of an algebra \(F_g(U,V,A)\) associated to an algebra \(A\), a left \(A\)-module \(U\), a right \(A\)-module \(V\), and a bilinear form \(g\colon U\times V\to A\). It is proved that any decomposition \(U=\bigoplus_{i=0}^nU_i\), \(V=\bigoplus_{i=0}^nV_i\), satisfying certain conditions induces a finite \(\mathbb{Z}\)-grading on \(F_g(U,V,A)\). Conversely, if \(R\) is a simple algebra with a finite \(\mathbb{Z}\)-grading, then \(R\) is graded isomorphic to some \(F_g(U,V,A)\).

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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