In this paper, we extend Mosic’s result for idempotency in associative rings to power-associative rings. We provide a necessary and sufficient condition for idempotency and give some examples.

In this article we provide a necessary and sufficient condition for idempotency in power-associative rings, hence extending Mosic’s result in [5]. Mosic gives the relation between idempotent and tripotent elements in an associative ring R, generalizing the result on matrices by Trenkler and Baksalary [8]. Namely, for any x ∈ R, where 2, 3 are invertible, x is idempotent if and only if x is tripotent and 1 − x is tripotent or 1 + x is invertible.

Full Article: https://rsmams.org/download/articles/1_13_1_1129180307_DOI%2010.56827JRSMMS.2025.1301.5.pdf