Biswas, R.Roy, F.2026-02-042024Linear Algebra and Its Applications, 2024, 689, , pp. 155-175243795https://doi.org/10.1016/j.laa.2024.02.024https://idr.nitk.ac.in/handle/123456789/21132For 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−2p≠(pq)m−1p for some m(≥2)∈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 αp+q under two different assumptions, (pq)m−1=(pq)m and λ(pq)m−1=(pq)m, where α is a non-zero and λ is a non-unit real or complex number. © 2024 Elsevier Inc.Associative algebrasConditionDrazin inverseDrazin inversionFinite-dimensional algebrasGroup inversionIdempotentInvertibilityTightly-coupledUpper BoundLinear algebra