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 $\mathfrak{A}=M_n(\mathbb{F}^σ[\mathsf{H}])$ with a canonical elementary $\mathsf{G}$-grading, where $\mathsf{H}$ is a finite abelian subgroup of $\mathsf{G}$ and $σ\in\mathsf{Z}^2(\mathsf{H},\mathbb{F}^*)$, 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.