Rings whose elements are represented by at most three commuting indempotents

Abstract

We completely characterize up to an isomorphism those rings whose elements are expressed as the sum of two, respectively three, commuting idempotents or are minus an idempotent. This strengthens well-known joint results in the subject due to Hirano-Tominaga (Bull. Austral. Math. Soc., 1988), Ahn-Anderson (Rocky Mount. J. Math., 2006), Danchev-McGovern (J. Algebra, 2015), Ying et al. (Can. Math. Bull., 2016), as well as own results established by Danchev (Bull. Iran. Math. Soc., 2019) and (Boll. Un. Mat. Ital., 2019).