Drazin and group invertibility in algebras spanned by two idempotents

For two given idempotents p and q from an associative algebra A, in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (pq)(m-1) = (pq)(m) but (pq)(m-2) p not equal (pq)(m-1)p for some m(>= 2) is an element of N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the 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), where alpha is a nonzero and lambda is a non-unit real or complex number. (c) 2024 Elsevier Inc. All rights reserved.