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Let $A$ be a unitary ring and let $(\mathbf{I(A),\subseteq })$ be the lattice of ideals of the ring $A.$ In this article we will study the property of the lattice $(\mathbf{I(A),\subseteq})$ to be Noetherian or not, for various types of rings $A$. In the last section of the article we study certain rings that are not Boolean rings, but all their ideals are idempotent.

Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $\xi_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(\xi_{n}\right)$, when $n$$\in$$\left\{3,4,6,7,8,9,11,12\right\}$ and also we obtain a characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(\xi_{n}\right)$, when $n$$\in$$\left\{5,10\right\}$. In the last section of this article we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(\xi_{n}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $n=p^{k},$ with $p$ is a prime integer, $p\equiv 3$ (mod $4$) and $k$ is a positive integer.

The aim of this paper is to study certain properties of the Kullback-Leibler distance between two positive integer numbers or between two ideals. We present some results related the entropy of a positive integer number and the divergence of two numbers. We also study the entropy of some types of ideals and the divergence of two ideals. Finally, we find some inequalities, involving the entropy H of an exponential divisor of a positive integer, respectively the entropy H of an exponential divisor of an ideal., Comment: arXiv admin note: substantial text overlap with arXiv:2210.12149

In this article we find some properties of certain types of entropies of a natural number. Also, regarding the entropy H of a natural number, introduced by Minculete and Pozna, we generalize this notion for ideals and we find some of its properties. In the last section we find some inequalities, involving the entropy H of an exponential divisor of a positive integer, respectively the entropy H of an exponential divisor of an ideal., 10 pages

Let $p$ be an odd prime, let $K=\mathbb{Q}(\epsilon)$ where $\epsilon$ is a primitive cubic root of unity, and let $L$ be the Kummer field $\mathbb{Q}\left(\epsilon, \sqrt[3]{\alpha}\right)$. In this paper we obtain a characterization of the splitting behavior of the symbol algebras $\left( \frac{\alpha ,p}{K,\epsilon }\right)$ and $\left( \frac{\alpha ,p^{h_{p}}}{K,\epsilon}\right)$, where $h_{p}$ is the order in the class group $Cl\left(L\right)$ of a prime ideal of $\mathcal{O}_L$ which divides $p\mathcal{O}_L.$

In this paper, we generalize the arithmetic functions $��$ and $��^{\left(e\right)}$ in algebraic number fields and we find some properties of these functions.

Let $d$ and $m$ be two distinct squarefree integers and $\mathcal{O}_K$ the ring of integers of the quadratic field $K=\mathbb{Q}(\sqrt{d})$. Denote by $ H_K(\alpha, m)$ a quaternion algebra over $K$, where $\alpha\in \mathcal{O}_K$. In this paper we give necessary and sufficient conditions for $ H_K(\alpha, m)$ to split over $K$ for some values of $\alpha$, and we obtain a complete characterization of division quaternion algebras $ H_K(\alpha, m)$ over $K$ whenever $\alpha$ and $m$ are two distinct positive prime integers. Examples are given involving prime Fibonacci numbers., Comment: arXiv admin note: substantial text overlap with arXiv:1802.08185

Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of quaternion division algebras H_K(p,q) over some dihedral fields K., This is a preliminary form of the article. Categories: number theory; rings and algebras

{\small In this paper we define and we study properties of} $\left(l,1,p+2q,q\cdot l\right) -$ {\small numbers,} $\left(l,1,p+2q,q\cdot l\right) -$ {\small quaternions,} $\left(l,1,p+2q,q\cdot l\right) -$ {\small symbol elements. Finally, we obtain an algebraic structure with these elements.}

In this paper we determine sufficient conditions for a quaternion algebra to split over a quadratic field. In the last section of the paper, we find a class of division symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq 3$) over a $p-$ adic field or over a cyclotomic field.

{\small In this paper, we find a class of division quaternion algebras over the field }$\mathbb{Q}\left( i\right) ${\small \ and a class of division symbol algebras over a cyclotomic field.}, 12 pages

In this paper we will study some properties of the matrix representations of symbol algebras of degree three, we study some equations with coefficients in these algebras, we find an octonion algebra in a symbol algebra of degree three, we define the Fibonacci symbol elements and we give some properties of them., Comment: improved version of the version from April 2012(final version)

In this paper, we obtain some properties of the symbol algebras, starting from their connections with the quaternion and cyclic algebras over a field $K_{p},$% where $K$ is an algebraic number field, $p$ is a prime in $K$ and $K_{p}$ is the completion of $K$ with respect to $p-$ adic valuation, in the case when $K=\Bbb{Q}(\varepsilon), \varepsilon ^{3}=1,\varepsilon \neq 1.$, This paper was accepted to publish in Carpathian Journal of Mathematics

In this paper we study the Diophantine equation $x^{4}-q^{4}=py^{5},$ with the following conditions: $p$ and $q$ are different prime natural numbers, $y$ is not divisible with $p$, $p\equiv3$ (mod20), $q\equiv4$ (mod5), $\overline{p}$ is a generator of the group $(U(\textbf{Z}_{q^{4}}),\cdot)$, $(x,y)=1$, 2 is a 5-power residue mod $q$., This paper was accepted for publication in Italian Journal of Pure and Applied Mathematics

We give a necessary condition for the existence of solutions of the Diophantine equation $p=x^{q}+ry^{q},$ with $p$, $q$, $r$ distinct odd prime natural numbers., Comment: Preliminary version

In this paper, we prove a theorem about the integer solutions to the Diophantine equation $x^{4}-q^{4}=py^{r}$, extending previous work of K.Gy\H ory, and F.Luca and A.Togbe, and of the author., Comment: 12 pages, Preliminary version