Year: 2020
Type: preprint
Abstract: Idempotent elements are a well-studied part of ring theory, with several identities of the idempotents in $\mathbb{Z}/m\mathbb{Z}$ already known. Although the idempotents are not closed under addition, there are still interesting additive identities that can be derived and used. In this paper, we give several new identities on idempotents in $\mathbb{Z}/ m\mathbb{Z}$. We relate finite sublattices over $\mathbb{Z}/ k\mathbb{Z}$ for all integers $k$ to an infinite lattice that is embedded in the divisibility lattice on $\mathbb{N}$ and to each other as sublattices of this infinite lattice. Using this relation, we generalize several identities on idempotents in $\mathbb{Z}/m\mathbb{Z}$ to those involving idempotents related to these finite sublattices. Finally, as an application of the above idempotent identities, we derive an algorithm for calculating modular exponentiation over $\mathbb{Z}/ m\mathbb{Z}$.
Source: arXiv (Cornell University)
Authors: Isham, Kelly, Monroe, Laura
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3022523995