Drazin and group invertibility in algebras spanned by two idempotents

Abstract

For two given idempotents $p\text{ and }q$ from an associative algebra $\mathcal{A},$ in this paper, we offer a comprehensive classification of algebras spanned by the idempotents $p\text{ and }q$. This classification is based on the condition that $p\text{ and }q$ are not tightly coupled and satisfies $(pq{)}^{m-1}=(pq{)}^m$ but $(pq{)}^{m-2}p\ne (pq{)}^{m-1}p$ for some $m(\ge 2)\in \mathbb{N}.$ Subsequently, we categorized all the group invertible elements and established an upper bound for Drazin index of any elements in these algebras spanned by $p,q$. Moreover, we formulate a new representation for the Drazin inverse of $(\alpha p+q)$ under two different assumptions, $(pq{)}^{m-1}=(pq{)}^m$ and $\lambda (pq{)}^{m-1}=(pq{)}^m,$ here $\alpha$ is a non-zero and $\lambda$ is a non-unit real or complex number.