Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Structure of the Generalized Group of Units   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.

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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: q-fin