On the Structure of the Generalized Group of Units

TL;DR

This paper investigates the structure of generalized groups of units in finite commutative rings, specifically for rings of integers modulo n, providing decompositions into cyclic groups and exploring their properties related to prime numbers.

Contribution

It offers a detailed decomposition of the generalized group of units for rfZ_n, extending previous work and linking the structure to Pratt Tree primes.

Findings

Decomposition of U^k(Z_n) into cyclic groups for any k

Conditions when these groups are boolean or trivial

Connection between group structure and Pratt Tree primes

Abstract

Let $R$ be a finite commutative ring with identity and $U(R)$ be its group of units. In 2005, El-Kassar and Chehade presented a ring structure for $U(R)$ and as a consequence they generalized this group of units to the generalized group of units $U^{k}\left( R\right) $ defined iteratively as the group of the units of $U^{k-1}(R)$, with $U^{1}\left( R\right) =U(R) $. In this paper, we examine the structure of this group, when $R=\mathbb{Z}_{n}.$ We find a decomposition of $U^{k}\left(\mathbb{Z}_{n}\right)$ as a direct product of cyclic groups for the general case of any $k$, and we study when these groups are boolean and trivial. We also show that this decomposition structure is directly related to the Pratt Tree primes.