Authors:Jose Maria GRAU; Celino MIGUEL, Antonio M. OLLER-MARCEN Abstract: In this paper we study the isomorphisms of generalized Hamilton quaternions $\Big(\frac{a,b}{R}\Big)$ where $R$ is a finite unital commutative ring of odd characteristic and $a,b \in R$. We obtain the number of non-isomorphic classes of generalized Hamilton quaternions in the case where $R$ is a principal ideal ring. This extends the case $R=\mathbb{Z}/n\mathbb{Z}$ where $n$ is an odd integer. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:El Hassane FLIOUET Abstract: A purely inseparable field extension $K$ of a field $k$ of characteristic $p\not=0$ is said to be $\omega_0$-generatedover $k$ if $K/k$ is not finitely generated, but $L/k$ is finitely generated for each proper intermediate field $L$.In 1986, Deveney solved the question posed by R. Gilmer and W. Heinzer, which consists in knowing if the lattice of intermediate fields of an $\omega_0$-generated field extension $K/k$ is necessarily linearly ordered under inclusion, by constructing an example of an $\omega_0$-generated field extension where $[k^{p^{-n}}\cap K: k]= p^{2n}$ for all positive integer $n$. This example has proved to be extremely useful in the construction of other examples of $\omega_0$-generated field extensions (of any finite irrationality degree).In this paper, we characterize the extensions of finite irrationality degree which are $\omega_0$-generated.In particular, in the case of unbounded irrationality degree, any modular extension of unbounded exponent contains a proper subfield of unbounded exponent over the ground field.Finally, we give a generalization, illustrated by an example, of the $\omega_0$-generated to include modular purely inseparable extensions of unbounded irrationality degree. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Antonino LEONARDIS; Gianfranco D'ATRI, Fabio CALDAROLA Abstract: Literature considers under the name "unimaginable numbers" anypositive integer going beyond any physical application. One of the most knownmethodologies to conceive such numbers is using hyper-operations, that is asequence of binary functions dened recursively starting from the usual chain:addition - multiplication - exponentiation. The most important notations torepresent such hyper-operations have been considered by Knuth, Goodstein,Ackermann and Conway as described in this work's introduction. Within thiswork we will give an axiomatic setup for this topic, and then try to nd on onehand other ways to represent unimaginable numbers, as well as on the otherhand applications to computer science, where the algorithmic nature of representations and the increased computation capabilities of computers give theperfect eld to develop further the topic, exploring some possibilities to effectively operate with such big numbers. In particular, we will give some axiomsand generalizations for the up-arrow notation and, considering a representation via rooted trees of the hereditary base-n notation, we will determine insome cases an effective bound related to "Goodstein sequences" using Knuthsnotation. Finally, we will also analyze some methods to compare big numbers,proving specically a theorem about approximation using scientic notationand a theorem on hyperoperation bounds for Steinhaus-Moser notation. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Marco Antonio Garcia MORALES; Lev GLEBSKY Abstract: Let $\Gamma$ be a group and $\mathscr{C}$ a class of groups endowed with bi-invariant metrics. We say that $\Gamma$ is $\mathscr{C}$-stable if every $\varepsilon$-homomorphism $\Gamma \rightarrow G$, $(G,d) \in \mathscr{C}$, is $\delta_\varepsilon$-close to a homomorphism, $\delta_\varepsilon\to 0$ when $\varepsilon\to 0$. If $\delta_\varepsilon < C \varepsilon$ for some $C$ we say that $\Gamma$ is $ \mathscr{C} $-stable with a linear rate. We say that $\Gamma$ has the property of defect diminishing if any asymptotic homomorphism can be changed a little to make errors essentially better. We show that the defect diminishing is equivalent to the stability with a linear rate. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Amr Ali Abdulkader AL-MAKTRY Abstract: Let RR be a commutative ring with unity 1≠01≠0. In this paper we introduce the definition of the first derivative property on the ideals of the polynomial ring R[x]R[x]. In particular, when RR is a finite local ring with principal maximal ideal m≠{0}m≠{0} of index of nilpotency ee, where 1<e≤ R/m +11≤e≤ R/m +1, we show that the null ideal consisting of polynomials inducing the zero function on RR satisfies this property. As an application, when RR is a finite local ring with null ideal satisfying this property, we prove that the stabilizer group of RR in the group of polynomial permutations on the ring R[x]/(x2)R[x]/(x2), is isomorphic to a certain factor group of the null ideal. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Mahnaz AHMADI; Ali S. JANFADA Abstract: We show that the quartic Diophantine equations $ax^4+by^4=cz^2$ has only trivial solution in the Gaussian integers for some particular choices of $a,b$ and $c$. Our strategy is by elliptic curves method. In fact, we exhibit two null-rank corresponding families of elliptic curves over Gaussian field. We also determine the torsion groups of both families. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Per BACK; Johan RICHTER Abstract: We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general ``twisting'' procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Ayman M. A. HOROUB; W. K. NICHOLSON Abstract: A ring is called left quasi-duo (left QD) if every maximal left ideal is aright ideal, and it is called I-finite if it contains no infinite orthogonalset of idempotents. It is shown that a ring is I-finite and left QD if andonly if it is a generalized upper-triangular matrix ring with all diagonalrings being division rings except the lower one, which is either a divisionring or it is I-finite, left QD and left `soclin' (left QDS). Here a ring iscalled left soclin if each simple left ideal is nilpotent. The left QDSrings are shown to be finite direct products of indecomposable left QDSrings, in each of which 1=f1+⋯+fm1=f1+⋯+fm where the fifi areorthogonal primitive idempotents, with fk≈flfk≈fl for all k,l,k,l,and ≈≈ is the block equivalence on {f1,…,fm}.{f1,…,fm}.A ring is shown to be left soclin if and only if every maximal left ideal isleft essential, if and only if the left socle is contained in the leftsingular ideal. These left soclin rings are proved to be a Morita invariantclass; and if a ring is semilocal and non-semisimple, then it is left soclinif and only if the Jacobson radical is essential as a left ideal..Left quasi-duo elements are defined for any ring and shown to constitute asubring containing the centre and the Jacobson radical of the ring. The`width' of any left QD ring is defined and applied to characterize thesemilocal left QD rings, and to clarify the semiperfect case.. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Nguyen Minh TRI Abstract: In this paper, we show that if $M$ is a non-zero Artinian $R$-module and $\underline{x}:=x_1,\ldots,x_n$ is an $M$-coregular sequence, then $x_1,\ldots,x_n$ is a $D(H_n^{\underline{x}}(M))$-coregular sequence. Moreover, if $R$ is complete with respect to $I$-adic topology and $d=\mathrm{Ndim} M$, then $\dim H^I_d(M) \le d$ and $\mathrm{depth} H_I^d(M)\ge \min\{{2, d}\}$ whenever $H^I_d(M) \ne 0.$ PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Samir BELHADJ; Mouloud GOUBİ Abstract: The present work is focused on the study of a cotangent sumassociated to the zeros of the Estermann zeta function and Riemannzeta function. We use Bell polynomials and generating functionsapproach to give arithmetical proof of its Dirichlet seriesdifferent from that given by M. Th. Rassias. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Ashkan NIKSERESHT Abstract: Let ΔΔ be a simplicial complex, IΔIΔ its Stanley-Reisner ideal and R=K[Δ]R=K[Δ] its Stanley-Reisner ring over a field KK. In 2018, the author introduced the squarefree zero-divisor graph of RR, denoted by Γsf(R)Γsf(R), and proved that if ΔΔ and Δ′Δ′ are two simplicial complexes, then the graphs Γsf(K[Δ])Γsf(K[Δ]) and Γsf(K[Δ′])Γsf(K[Δ′]) are isomorphic if and only if the rings K[Δ]K[Δ] and K[Δ′]K[Δ′] are isomorphic. Here we derive some algebraic properties of RR using combinatorial properties of Γsf(R)Γsf(R). In particular, we state combinatorial conditions on Γsf(R)Γsf(R) which are necessary or sufficient for RR to be Cohen-Macaulay. Moreover, we investigate when Γsf(R)Γsf(R) is in some well-known classes of graphs and show that in these cases, IΔIΔ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γsf(R)Γsf(R) and their algebraic interpretations. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Feroz SIDDIQUE Abstract: We show that a ring $R$ has stable range one if and only if everyleft unit lifts modulo every left principal ideal. We also showthat a left quasi-morphic ring has stable range one if and only ifit is left uniquely generated. Thus we answer in the affirmativethe two questions raised by W. K. Nicholson. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:H. ANSARI-TOROGHY; S. S. POURMORTAZAVİ Abstract: Let $R$ be a commutative ring with identity, $S$ a multiplicatively closed subset of $R$, and $M$ be an $R$-module. In this paper, we study and investigate some properties of $S$-primary submodules of $M$. Among the other results, it is shown that this class of modules contains the family of primary (resp. $S$-prime) submodules properly. PubDate: Mon, 17 Jan 2022 00:00:00 +030

Authors:Erwin CERDA-LEON; Hugo RINCON-MEJIA Abstract: We introduce some new lattices of classes of modules with respect to appropriate preradicals. We introduce some concepts associated with these lattices, such as the $\sigma$-semiartinian rings, the $\sigma$-retractable modules, the $\sigma$-$V$-rings, the $\sigma$-max rings. We continue to study $\sigma$-torsion theories, $\sigma$-open classes, $\sigma$-stable classes. We prove some theorems that extend some known results. Our results fall into well known situations when the preradical $\sigma$ is chosen as the identity preradical. PubDate: Mon, 17 Jan 2022 00:00:00 +030