Arithmetic of idempotents in $\mathbb{Z}/m \mathbb{Z}$

Abstract

Idempotent elements are a well-studied part of ring theory, with several identities of the idempotents in Z/mZ 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 Z/mZ. We relate finite sublattices over Z/kZ for all integers k to an infinite lattice that is embedded in the divisibility lattice on N and to each other as sublattices of this infinite lattice. Using this relation, we generalize several identities on idempotents in Z/mZ 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 Z/mZ.

Idempotents of $2\times 2$ matrix rings over rings of formal power series

“…See also the recent preprint of Nguyen [12] and the references there for further developments in this direction. We shall mention also the paper by Ánh, Birkenmeier and van Wyk [1] where the authors mimic the behavior of idempotents in matrix rings in a more general setup and the following three papers which are related to the present project: by Birkenmeier, Kim and Park [4] and Kanwar, Leroy and Matczuk [10] for relations between the idempotents of A, A[X] and A[[X]] where X is a finite set of variables and by Isham and Monroe [8] for the properties of the idempotents in Z n .…”