Irreducible graded bimodules over algebras and a Pierce decomposition of the Jacobson radical

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let G be a group, F an algebraically closed field with char(F)=0, A a finite dimensional G-graded associative F -algebra and M a G-graded unitary A -bimodule. We proved that if A=Mn(F sigma[H]) with an elementary-canonical G-grading, where H is a finite abelian subgroup of G and sigma is an element of Z2(H,F*) , then M being irreducible graded implies that there exists a nonzero homogeneous element w is an element of M satisfying M=Bw and Bw=wB . Another result we proved generalizes the last one: if G is abelian, A is simple graded and M is finitely generated, then there exist nonzero homogeneous elements w1,w2,& mldr;,wn is an element of M such that M=Aw1 circle plus Aw2 circle plus & ctdot;circle plus Awn , where wiA=Awi not equal 0 for all i=1,2,& mldr;,n , and each Awi is irreducible. The elements wi 's are associated with the irreducible characters of G. We also describe graded bimodules over graded semisimple algebras. And we finish by presenting a Pierce decomposition of the graded Jacobson radical of any finite dimensional F -algebra with a G-grading.