On (m, k) -type elements in the ring of integers modulo n

Abstract

An element a in a ring R is said to be of (m, k)-type if a m = a k where m and k are positive integers with m > k ≥ 1. Let Xn(m, k) be the set of all (m, k)-type elements, X * n(m, k) be the set of all nonzero (m, k)-type elements, and Sn(m, k) be the set of all nonunit (m, k)-type elements in the ring of integers modulo n. In this paper, we study the algebraic structures of Xn(m, k), X * n(m, k) and Sn(m, k) and characterize all values of n, m, and k for which Xn(m, k) and Sn(m, k) are cyclic semigroups and X * n(m, k) is a cyclic group.

Keywords

• (m
• k)-type elements
• ring of integers modulo n
• m-potents
• cyclic semigroup
• cyclic group