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Let $R$ be a commutative ring with a collection of ideals $\{ N_1, N_2, \dots, N_{k-1}\}$ satisfying certain conditions, properties of the set of invertible quadratic residues of the ring $R$ are described in terms of properties of the set of invertible quadratic residues of the quotient ring $R/N_1$., Comment: arXiv admin note: text overlap with arXiv:2012.06949, arXiv:1911.07743

Considering $\mathbb{Z}_n$ the ring of integers modulo $n$, the classical Fermat-Euler theorem establishes the existence of a specific natural number $\varphi(n)$ satisfying the following property: $ x^{\varphi(n)}=1%\hspace{1.0cm}\text{for all}\hspace{0.2cm}x\in \mathbb{Z}_n^*, $ for all $x$ belonging to the group of units of $\mathbb{Z}_n$. In this manuscript, this result is extended to a class of rings that satisfies some mild conditions., Comment: arXiv admin note: text overlap with arXiv:1911.07743

Let $R$ be an associative ring with identity and let $N$ be a nil ideal of $R$. It is shown that units of $R/N$ can be lifted to units in $R$. Under some mild conditions on the ring, a procedure is given to determine those lifted units in a recursive way. As an application, the units of several classes of rings are determined, including: matrix rings, chain rings, and group rings where the ring is a chain ring. Numerical examples are given illustrating the main results., Comment: arXiv admin note: text overlap with arXiv:1904.12932 corrected typos

In the present work, a procedure for determining idempotents of a commutative ring having a sequence of ideals with certain properties is presented. As an application of this procedure, idempotent elements of various commutative rings are determined. Several examples are included illustrating the main results., Comment: In this version, changes were made in the title, abstract and introduction to make the main theorem of the article clearer. No changes were made to the results presented in the previous version

In a former paper the authors introduced two new systematic authentication codes based on the Gray map over a Galois ring. In this paper, it is proved the one-to-one onto correspondence between keys and encoding maps for the second introduced authentication code.