Idempotents and tridempotents in ℤn, ℤn[i], and ℤn[ω]

Gallian, Joseph; Mattson, Blake; Moulton, David Petrie

doi:10.2140/involve.2026.19.181

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Idempotents and tridempotents in $\mathbb{Z}_n$, $\mathbb{Z}_n{[}i{]}$, and $\mathbb{Z}_n{[}\omega{]}$

Joseph Gallian, Blake Mattson and David Petrie Moulton

Vol. 19 (2026), No. 2, 181–206

DOI: 10.2140/involve.2026.19.181

Abstract

We determine the number and the exact form of the involutions ( $a^{2} = 1$ ), idempotents ( $a^{2} = a$ ), and tridempotents ( $a^{3} = a$ ) in $ℤ_{n}$ , in the ring of Gaussian integers modulo $n$ , $ℤ_{n} [i]$ , and in the ring of Eisenstein integers modulo $n$ , $ℤ_{n} [ω]$ , where $ω$ is a primitive cube root of unity. We also provide two useful characterizations of tridempotents.

Keywords

idempotents, involutions, tridempotents, Gaussian integers, Eisenstein integers

Mathematical Subject Classification

Primary: 11A05, 11R11

Milestones

Received: 26 March 2023

Revised: 13 October 2024

Accepted: 23 October 2024

Published: 8 March 2026

Communicated by Scott T. Chapman

Authors

Joseph Gallian
Department of Mathematics and Statistics University of Minnesota Duluth Duluth, MN United States

Blake Mattson
Department of Mathematics University of Iowa Iowa City, IA United States

David Petrie Moulton
Google Sunnyvale San Francisco, CA United States

Vol. 19, No. 2, 2026