Representability of matrices over commutative rings as sums of two potent matrices (Q6641626)

Continuing in a natural way the results from [\textit{A. Abyzov} et al., Turk. J. Math. 48, No. 5, 817--839 (2024; Zbl 07926319)], obtained by the present two authors together with Cohen and Danchev, the work under review aims to deal with the more general problem of finding certain representabilities of square matrices over commutative rings as the sum of two potent matrices sometime of possibly different indices of potentness.\N\NThe major theorems are Propositions 7 and 14 as well as Theorems 16, 18, 19, 20, 21, 22 and 25, respectively. All of them are equipped with correct and complete proofs which use a non-standard machinery of different aspects.\N\NMoreover, three intriguing conjectures, namely Conjecture 1, Conjecture 2 and Conjecture 23, are also raised in the paper that, hopefully, will stimulate a further intensive exploration of the current subject.\N\NThe work is finished by a difficult open question, which seems to be insurmountable in its generality at this stage.\N\NThe article is well motivated and written, the established results are non-trivial and interesting, so that the presented work contributes significantly on the topic and the interested reader will definitely enjoy reading its content.