Periodic and potent elements

• DOI: 10.1142/s0219498825503323
• Publication Date: 2024-06-17T14:04:26Z
• Authors: Guanglin Ma; André Leroy; Mehrdad Nasernejad
• Keywords: 0102 computer and information sciences; 01 natural sciences

In this paper, our main aim is to study periodic and potent elements of a ring. We specially study periodic elements of graded rings and generalize some classical results related to idempotent of polynomial rings. We show that a (von Neumann) quasi-inverse of a potent element is a root of unity. We study the isomorphism of potent elements and analyze some closure properties of the set [Formula: see text] of potent elements of a ring [Formula: see text]. The potent elements of the endomorphism ring of a Fitting module are described, and we apply this to matrices over division rings. In the case of matrices over finite fields, we connect features of potent elements with the exponent of their minimal polynomials.