Structure theorem for generalized corner rings

TL;DR

This paper characterizes the structure of generalized corner rings in unital rings when the element's square has finite rank, extending understanding of ring substructures with a necessary and optimal assumption.

Contribution

It provides a complete description of the structure of $aRa$ under finite rank conditions on $a^2$, utilizing recent results on element ranks in rings.

Findings

Complete structural description of $aRa$ when $a^2$ has finite rank.

Demonstrates the necessity and optimality of the finite rank assumption.

Provides an example illustrating the limits of the main result.

Abstract

We apply recent results on the rank of elements of rings to study the structure of generalized corner rings $aRa$, where $R$ is a unital ring and $a$ an element of $R$. We give a complete description of the structure of $aRa$ when $a^2$ has finite rank and provide an example to show that this assumption is necessary and optimal.