## PI-equivalência em álgebras graduadas simples

##### Abstract

This work aims to give a description, under certain hypothesis, of the graded simple algebras and prove that they are determined by their graded identities. For this, we study the papers [3] and [19]. More precisely we will show the following: Let G be a group, F an algebraically closed eld, and R = L g2G Rg a finite dimensional G-graded F-algebra such that the order of each finite subgroup of G is invertible in F. Then R is a G-graded simple algebra if and only if R is isomorphic, as graded algebra, to the tensor product C = Mn(F) F [H], where H is a nite subgroup of G, is a 2-cocycle in H, Mn(F) has an elementary G-grading, F [H] has a canonical grading and C has an induced G-grading by the tensor product. Based on this result, admitting the same assumptions and adding that G is an abelian group, we prove that two graded simple algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras.