On the ideal structure of the tensor product of nearly simple algebras

TL;DR

This paper investigates the ideal structure of tensor products of nearly simple algebras, aiming to characterize when all non-trivial ideals are generated by the ideals of the factors.

Contribution

It provides a characterization of when the non-trivial ideals of tensor products of nearly simple algebras are exactly those generated by the ideals of the individual algebras.

Findings

Identifies conditions under which all non-trivial ideals of the tensor product are of specific forms.

Characterizes the ideal structure for tensor products of nearly simple algebras.

Provides criteria for the ideal structure to be generated by the ideals of the factors.

Abstract

We define a unital algebra $A$ over a field $\mathbb{F}$ to be nearly simple if $A$ contains a unique non-trivial ideal $I_A$ such that $I_A^2 \neq \{0\}$. If $A$ and $B$ are two nearly simple algebras, we consider the ideal structure of their tensor product $A \otimes B$. The obvious non-trivial ideals of $A \otimes B$ are: $$I_A \otimes I_B, \quad I_A \otimes B, \quad A \otimes I_B, \quad \mbox{and} \quad I_A \otimes B + A \otimes I_B.$$ The purpose of this paper is to characterize when are all non-trivial ideals of $A \otimes B$ of the above form.