On the Structure of the Generalized Group of Units

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.