Irreducible Graded Bimodules over Algebras and a Pierce Decomposition of the Jacobson Radical

Abstract: 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 $\mathsf{G}$ be a group, $\mathbb{F}$ an algebraically closed field with $\mathsf{char}(\mathbb{F})=0$, $\mathfrak{A}$ a finite dimensional $\mathsf{G}$-graded associative $\mathbb{F}$-algebra and $\mathsf{M}$ a $\mathsf{G}$-graded unitary $\mathfrak{A}$-bimodule. We proved that if $\mathfrak{A}=M_n(\mathbb{F}^\sigma[\mathsf{H}])$ with a canonical elementary $\mathsf{G}$-grading, where $\mathsf{H}$ is a finite abelian subgroup of $\mathsf{G}$ and $\sigma\in\mathsf{Z}^2(\mathsf{H},\mathbb{F}^*)$, then $\mathsf{M}$ being irreducible graded implies that there exists a nonzero homogeneous element $w\in\mathsf{M}$ satisfying $\mathsf{M}=\mathfrak{B}w$ and $\mathfrak{B} w= w\mathfrak{B}$. Another result we proved generalizes the last one: if $\mathsf{G}$ is abelian, $\mathfrak{A}$ is simple graded and $\mathsf{M}$ is finitely generated, then there exist nonzero homogeneous elements $w_1, w_2,\dots,w_n\in\mathsf{M}$ such that \begin{equation}\nonumber \mathsf{M}=\mathfrak{A} w_1\oplus\mathfrak{A} w_2\oplus \cdots \oplus \mathfrak{A} w_n \ , \end{equation} where $w_i \mathfrak{A}=\mathfrak{A} w_i\neq0$ for all $i=1, 2,\dots,n$, and each $\mathfrak{A} w_i$ is irreducible. The elements $w_i$'s are associated with the irreducible characters of $\mathsf{G}$. 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 $\mathbb{F}$-algebra with a $\mathsf{G}$-grading.