Finite Z-gradings of simple associative algebras

Abstract

We show that every finite Z-grading of a simple associative algebra A comes from a Peirce decomposition induced by a complete system of orthogonal idempotents lying in the maximal left quotient algebra of A (which coincides with the graded maximal left quotient algebra of A). Moreover, a nontrivial 3-grading can be found. This grading provides 3-gradings in simple M-graded Lie algebras. Some consequences are obtained for left nonsingular algebras with a finite Z-grading.