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 G be a group, $\mathbb{F}$ an algebraically closed field with $\mathsf{\text{char}}(\mathbb{F})=0,\mathfrak{A}$a finite dimensional G-graded associative $\mathbb{F}$-algebra and M a G-graded unitary $\mathfrak{A}$-bimodule. We proved that if $\mathfrak{A}=M_n({\mathbb{F}}^{\sigma }[\mathsf{H}])$ with an elementary-canonical G-grading, where H is a finite abelian subgroup of G and $\sigma \in {\mathsf{Z}}^2(\mathsf{H},{\mathbb{F}}^{*})$, then 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 G is abelian, $\mathfrak{A}$ is simple graded and M is finitely generated, then there exist nonzero homogeneous elements $w_1,w_2,\dots ,w_n\in \mathsf{M}$ such that $\mathsf{M}=\mathfrak{A}{w}_1\oplus \mathfrak{A}{w}_2\oplus \cdots \oplus \mathfrak{A}{w}_n,$ where $w_i\mathfrak{A}=\mathfrak{A}{w}_i\ne 0$ 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 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 G-grading.

Keywords:

• G-graded bimodule
• G-irreducible bimodule
• G-simple bimodule
• Jacobson radical
• Pierce decomposition
• Specht’s problem
• weak G-artinian bimodule
• weak G-noetherian bimodule

2020 Mathematics Subject Classification:

• Primary: 16D20
• Secondary: 16D70
• 16W50
• 16P20
• 16P40

Acknowledgments

The work was carried out when the first author was a Professor at the University of Brasília, between 2022 and 2023. The authors are thankful to the article’s referees for the careful reading and very useful stylistic recommendations.

Notes

1 An $(\mathfrak{A},\tilde{\mathfrak{A}})$-bimodule M is a left $\mathfrak{A}$-module and a right $\tilde{\mathfrak{A}}$-module that satisfies $a(\mathit{\text{mb}})=(\mathit{\text{am}})b$ for any $a\in \mathfrak{A},b\in \tilde{\mathfrak{A}}$ and $m\in \mathsf{M}$.

2 An algebra $\mathfrak{A}$ is called an alternative algebra if satisfies $(\mathit{\text{xx}})y=x(\mathit{\text{xy}})$ and $(\mathit{\text{xy}})y=x(\mathit{\text{yy}})$ for any $x,y\in \mathfrak{A}$.

3 The Wedderburn-Malcev Theorem (see [12], Theorem 72.19, p. 491), which is a generalization made by Malcev, in [33], of one of Wedderburn’s Theorem, states that for any finite dimensional algebra over a field $\mathbb{F}$ such that $\mathfrak{A}/\mathsf{J}(\mathfrak{A})$ is separable, there exists a unique (up to the isomorphism) maximal semisimple subalgebra $\mathfrak{B}$ of $\mathfrak{A}$ satisfying $\mathfrak{A}=\mathfrak{B}\oplus \mathsf{J}(\mathfrak{A})$.

4 A graded variety generated by $S\subset \mathbb{F}〈X^{\mathsf{G}}〉$, denoted by $\mathsf{\text{va}}{\mathsf{r}}^{\mathsf{G}}(S)$, is the class of graded associative algebras that satisfy any $g\in S$, i.e. a G-graded algebra $\mathfrak{A}$ belongs to $\mathsf{\text{va}}{\mathsf{r}}^{\mathsf{G}}(S)$ iff $g{\equiv }_{\mathsf{G}}0$ in $\mathfrak{A}$ for any $g\in S$. For more details about (graded) varieties of (graded) algebras, see [15], Chapter 2, or [20], Chapter 1.

5 Specht’s Problem was purposed in [42] by W. Specht (1950), and it can be formulated by the following question: given any algebra $\mathfrak{A}$, is any set of polynomial identities of $\mathfrak{A}$a consequence of a finite number of identities of $\mathfrak{A}$? For more details about Specht’s Problem, see [8]. We also suggest the works [2, 28, 29, 37], and [11].

6 The Grassmann envelope of a $(\mathsf{G}\times {\mathbb{Z}}_2)$-graded algebra $\mathfrak{A}$ is given by ${\mathsf{E}}^{\mathsf{G}}(\mathfrak{A})=({\mathfrak{A}}_0\otimes {\mathsf{E}}_0)\oplus ({\mathfrak{A}}_1\otimes {\mathsf{E}}_1)$, which is naturally $\mathsf{G}\times {\mathbb{Z}}_2$-graded and G-graded. Here, $\mathsf{E}={\mathsf{E}}_0\oplus {\mathsf{E}}_1$ is an infinitely generated non-unitary Grassmann algebra, i.e. $\mathsf{E}=〈e_1,e_2,e_3,\dots |e_i{e}_j=-e_j{e}_i,\forall i,j〉$ is ${\mathbb{Z}}_2$-graded with ${\mathsf{E}}_0=\mathsf{\text{spa}}{\mathsf{n}}_{\mathbb{F}}\{e_{i_1}{e}_{i_2}\cdots e_{i_n}:n\text{ is even}\}$, and ${\mathsf{E}}_1=\mathsf{\text{spa}}{\mathsf{n}}_{\mathbb{F}}\{e_{j_1}{e}_{j_2}\cdots e_{j_m}:m\text{ is odd}\}$ (see Section 3.7 in [20], pp. 80–83).

Additional information

Funding

The first author was partially supported by Paraíba State Research Foundation (FAPESQ), Grant #2023/2158.