Journal of Algebra and Its Applications

It is known that quaternion polynomials may have spherical zeros and isolated left and right zeros. These zeros along with appropriately defined multiplicities form the zero structure of a polynomial which can be alternatively described in terms of left and right spherical divisors of a polynomial as well as in terms of its left and right indecomposable divisors. These alternative descriptions are used to construct a polynomial with prescribed zero structure and more generally, to construct the least common multiple of given polynomials. Similar questions are discussed in the context of quaternion power series and particularly, finite Blaschke products.

We study the semigroup structure on the set $C(A)$ of conjugacy classes of left ideals of a finite-dimensional algebra $A$ over an algebraically closed field $\mathrm{\text{K}}$, equipped with the natural multiplication inherited from $A$, and the structure of the contracted semigroup algebra ${\mathrm{\text{K}}}_0[C(A)]$. It is shown that $C(A)$ has a finite chain of ideals with either nilpotent or completely $0$-simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver $Q_A$ of $A$ is proved to be a subquiver of $Q_{{\mathrm{\text{K}}}_0[C(A)]}$, if $C(A)$ is finite. Moreover, in this case, the structure of $C(A)$ determines, up to isomorphism, the structure of the algebra $A$ modulo its Jacobson radical. Combining these results we show that if the semigroup $C(A)$ is finite, then it determines the structure of any (not necessarily basic) triangular algebra $A$ which admits a normed presentation.

Let $R$ be a ring, $(S,\le )$ a strictly ordered monoid and $\omega :S\to \mathrm{\text{End}}(R)$ a monoid homomorphism. The skew generalized power series ring $R[[S,\omega ]]$ is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. In this paper, we consider the problem of determining when $f\in R[[S,\omega ]]$ is nilpotent in $R[[S,\omega ]]$. We study various annihilator properties and a variety of conditions and related properties that the skew generalized power series $R[[S,\omega ]]$ inherits from $R$. We also introduce and study the $(S,\omega )$-McCoy condition on $R$, a generalization of the standard McCoy condition from polynomials to skew generalized power series. We resolve the structure of $(S,\omega )$-McCoy rings and obtain various necessary or sufficient conditions for a ring to be $(S,\omega )$-McCoy. As particular cases of our general results we obtain several new theorems on the McCoy condition. Moreover various examples of $(S,\omega )$-McCoy rings are provided.

This is a study of rings whose cyclic modules are $D3$-modules with applications to rings whose cyclics are quasi-discrete and, respectively, discrete.

Let $R\subset S$ be an extension of integral domains, and let $R^{\ast }$ be the integral closure of $R$ in $S$. The main purpose of this paper is to study $[R,S]$, the set of intermediate rings between $R$ and $S$. As a main tool, we establish an explicit description of any intermediate ring in terms of localizations of $R$ (or $R^{\ast }$). This study effectively enables us to characterize the minimal extensions in $[R,S]$. We also prove a special chain theorem concerning the length of an arbitrary maximal chain in $[R,S]$.

In this work, $p$-dimensional point-lattices are constructed from a number field $L$, where $L/\mathbb{Q}$ is a cyclic extension of degree $p$, an odd unramified prime in $L/\mathbb{Q}$. Families of lattices whose packing densities asymptotically approach those of ${\mathrm{\Lambda }}_p$, $p=3,5,7$, are explicitly exhibited.

Recently, Xiang and Ouyang defined a (commutative unital) ring $R$ to be Nil${}_{*}$-coherent if each finitely generated ideal of $R$ that is contained in Nil$(R)$ is a finitely presented $R$-module. We define and study Nil${}_{*}$-coherent modules and special Nil${}_{*}$-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil${}_{*}$-coherent ring and any special Nil${}_{*}$-coherent ring is a Nil${}_{*}$-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil${}_{*}$-coherent ring (regardless of whether it is coherent). Several examples of Nil${}_{*}$-coherent rings that are not special Nil${}_{*}$-coherent rings are obtained as byproducts of our study of the transfer of the Nil${}_{*}$-coherent and the special Nil${}_{*}$-coherent properties to trivial ring extensions and amalgamated algebras.

For a group $G$, let $\gamma (G)$ denote the number of conjugacy classes of non-nilpotent subgroups of $G$. We classify all finite groups $G$ with $\gamma (G)\le 3$

Given a $4$-dimensional vector subspace $U=\{f_0,\dots ,f_3\}$ of $H^0({\mathbb{P}}^1\times {\mathbb{P}}^1,{𝒪}(a,b))$, a tensor product surface, denoted by $X_U$, is the closure of the image of the rational map ${\lambda }_U:{\mathbb{P}}^1\times {\mathbb{P}}^1--\to {\mathbb{P}}^3$ determined by $U$. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of $X_U$ in ${\mathbb{P}}^3$. In this paper, we show that if $U\subseteq H^0({\mathbb{P}}^1\times {\mathbb{P}}^1,{𝒪}(a,1))$ has a finite set of $r$ basepoints in generic position, then the implicit equation of $X_U$ is determined by two syzygies of $I_U=〈f_0,\dots ,f_3〉$ in bidegrees $(a-\lceil \frac{r}{2}\rceil ,0)$ and $(a-\lfloor \frac{r}{2}\rfloor ,0)$. This result is proved by understanding the geometry of the basepoints of $U$ in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The proof techniques for the main theorem also apply when $U$ is basepoint free.

We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov $\mathrm{\Omega }$-words on a set $X$ and prove that the set of all the nonassociative Lyndon–Shirshov $\mathrm{\Omega }$-words forms a linear basis of the free Lie $\mathrm{\Omega }$-algebra on the set $X$. From this, we establish Gröbner–Shirshov bases theory for Lie $\mathrm{\Omega }$-algebras. As applications, we give Gröbner–Shirshov bases of a free $\lambda$-Rota–Baxter Lie algebra, of a free modified $\lambda$-Rota–Baxter Lie algebra, and of a free Nijenhuis Lie algebra and, then linear bases of these three algebras are obtained.

A prime ideal $𝔭$ is said to be strongly prime if whenever $𝔭$ contains an intersection of ideals, $𝔭$ contains one of the ideals in the intersection. A commutative ring with this property for every prime ideal is called strongly zero-dimensional. Some equivalent conditions are given and it is proved that a zero-dimensional ring is strongly zero-dimensional if and only if the ring is quasi-semi-local. A ring is called strongly $n$-regular if in each ideal $𝔞$, there is an element $a$ such that $x=ax$ for all $x\in {𝔞}^n$. Connections between the concepts strongly zero-dimensional and strongly $n$-regular are considered.

Let $G$ be a subgroup of the full automorphism group of a $2$-$(v,k,\lambda )$ symmetric design ${𝒟}$. If $G$ is flag-transitive and point-primitive, then Soc$(G)$ cannot be ${}^2{B}_2(q)$, ${}^3{D}_4(q)$, ${}^2{G}_2(q)$ and ${}^2{F}_4(q)$.

It is well known that the concept of left serial ring is a Morita invariant property and a theorem due to Nakayama and Skornyakov states that “for a ring $R$, all left $R$-modules are serial if and only if $R$ is an Artinian serial ring”. Most recently the notions of “prime uniserial modules” and “prime serial modules” have been introduced and studied by Behboodi and Fazelpour in [Prime uniserial modules and rings, submitted; Noetherian rings whose modules are prime serial, Algebras and Represent. Theory 19(4) (2016) 11 pp]. An $R$-module $M$ is called prime uniserial ( $\wp$-uniserial) if its prime submodules are linearly ordered with respect to inclusion, and an $R$-module $M$ is called prime serial ( $\wp$-serial) if $M$ is a direct sum of $\wp$-uniserial modules. In this paper, it is shown that the $\wp$-serial property is a Morita invariant property. Also, we study what happens if, in the above Nakayama–Skornyakov Theorem, instead of considering rings for which all modules are serial, we consider rings for which every $\wp$-serial module is serial. Let $R$ be Morita equivalent to a commutative ring $S$. It is shown that every $\wp$-uniserial left $R$-module is uniserial if and only if $R$ is a zero-dimensional arithmetic ring with $J(R)$ T-nilpotent. Moreover, if $S$ is Noetherian, then every $\wp$-serial left $R$-module is serial if and only if $R$ is serial ring with dim$(R)\le 1$.

A ring $R$ is central Armendariz if $f(x)=a_0+a_{1}x+\cdots +a_m{x}^m$ and $g(x)=b_0+b_{1}x$ $+\cdots +b_n{x}^n$ over $R$ satisfy $f(x)g(x)=0$ then all $a_i{b}_j$ are central. It is proved that if $R$ is a central Armendariz ring, then $f(x)g(x)=0$ implies that all $a_i{b}_j$ are in its prime radical.

A Cayley graph $\mathrm{\Gamma }=Cay(G,S)$ is said to be core-free if $G$ is core-free in some $X$ for $G\le X\le Aut\mathrm{\Gamma }$. A graph $\mathrm{\Gamma }$ is called $s$-regular if $Aut\mathrm{\Gamma }$ acts regularly on its $s$-arcs. It is shown in this paper that if $s\le 2$, then there exist no core-free tetravalent $s$-regular Cayley graphs; and for $s\ge 3$, every tetravalent $s$-regular Cayley graph is a normal cover of one of the three known core-free graphs. In particular, a characterization of tetravalent $s$-regular Cayley graphs is given.

In this note, we use fewer $p$-subgroups $H$ with the condition $O^p(G)\cap H⊴O^p(G)$ to investigate the structure of finite groups. We prove that for a fixed prime $p$, a given Sylow $p$-subgroup $P$ of a finite group $G$, and a power $d$ of $p$ dividing $\left|G\right|$ such that $p^2\le d<\left|P\right|$, if $H\cap O^p(G)$ is normal in $O^p(G)$ for all non-cyclic normal subgroups $H$ of $P$ with $\left|H\right|=d$, then either $G$ is $p$-supersoluble or else $|P\cap O^p(G)|>d$. This extends the main result of Guo and Isaacs (Conditions on $p$-subgroups implying $p$-nilpotence or $p$-supersolvability, Arch. Math. 105 (2015) 215–222). We also derive some applications of the above result which extend some known results.

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

Let $R$ be a discrete valuation ring with maximal ideal $𝔭$ and $S$ be the integral closure of $R$ in a finite separable extension $L$ of $K$. For a maximal ideal $𝔓$ of $S$, let ${\widehat{R}}_{𝔭},{Ŝ}_{𝔓}$ denote respectively the valuation rings of the completions of $K,L$ with respect to $𝔭,𝔓$. The discriminant satisfies a basic equality which says that $\mathrm{\text{disc}}(S/R){\widehat{R}}_{𝔭}={\prod }_{𝔓|𝔭}\mathrm{\text{disc}}({Ŝ}_{𝔓}/{\widehat{R}}_{𝔭})$. In this paper, we extend the above equality on replacing $R$ by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.

In this paper, we characterize the classification of simple maps from the plane to the plane given by J. H. Rieger, in terms of invariants. On the basis of this characterization we present an algorithm to classify the simple maps from the plane to the plane and also give its implementation in computer algebra system SINGULAR.

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.