Finite ZZ-gradings of simple associative algebras

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