Journal of Algebra and Its Applications

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring $D[x,\sigma ,\delta ]$. We extend results of the first author and Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a\in D$ and $f\in D[x,\sigma ,\delta ]$ satisfy $f(a)=a$, then $f^{\circ n}(a)=a$ for every formal power of $f$. More generally, we give a sufficient condition for a point $a$ to be $r$-periodic with respect to a polynomial $f$. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.

A module $M$ is said to be extending (Goldie extending) if for each submodule $X\le M,$ there exists a direct summand $D$ of $M$ such that $X$ is essential in $D$ (if for every submodule $X$ of $M$ there exists a direct summand $D$ of $M$ such that $X\cap D$ is essential in both $X$ and $D$). In this paper, a necessary and sufficient condition is obtained for a trivial Morita context ring to be right Goldie-extending.

In this paper, we study the structure of the skew polynomial ring $R=({𝔽}_p+u{𝔽}_p)[x;𝜃]$ and its quotient ring $R_n=R/〈x^n-1〉$, where $p$ is an odd prime number, $u^2=0$, and $𝜃(u)=-u$. We give an explicit structure of the ideals in $R$ and $R_n$ and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian.

Let $R$ be a noncommutative prime ring of characteristic different from $2$, with right Martindale quotient ring $Q_r$ and extended centroid $C$. Let $m,n\ge 1$ be fixed integers and $F$ a nonzero generalized skew derivation of $R$. In this paper, we investigate the set

The well-known notion of an extending module is closely linked to that of a Baer module. A right $R$-module $M$ is called extending if every submodule of $M$ is essential in a direct summand. On the other hand, a right $R$-module $M$ is called Baer if for all $N\le M$, $l_S(N){\le }^{\oplus }{}_{S}S$ where $S={\mathrm{\text{End}}}_R(M)$. In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc.  21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: “a module$M$ is${𝒦}$-nonsingular extending if and only if$M$ is${𝒦}$-cononsingular Baer”. $M_R$ is called ${𝒦}$-nonsingular if $\forall \phi \in S$ such that $\mathrm{\text{Ker}}\phi {\le }^{e}M$, $\phi =0$. Moreover, $M_R$ is called ${𝒦}$-cononsingular if for any $N\le M$ with $\phi N\ne 0$ for all $0\ne \phi \in S$, implies $N{\le }^{e}M$. In view of this result, every Baer module which happens to be ${𝒦}$-cononsingular will automatically become an extending module. In this paper, our main focus is the study of ${𝒦}$-cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of ${𝒦}$-cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of ${𝒦}$-cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman’s result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided.

In this paper, we shall study three types of Galois extensions $B$ which are Azumaya algebras over its center $C$. (1) ${{𝒜}}_1$, the class of Azumaya–Galois extensions; (2) ${{𝒜}}_2$, the class of Galois extensions $B$ of $B^G$ with Galois group $G$ such that $B^G$ is a separable $C^G$-algebra; and (3) ${{𝒜}}_3$, the class of Galois extensions which are Azumaya algebras over its center $C$. Clearly, ${{𝒜}}_1\subseteq {{𝒜}}_2\subseteq {{𝒜}}_3$. Examples are given to show the proper inclusion relationship ${{𝒜}}_1\subset {{𝒜}}_2\subset {{𝒜}}_3$, and equivalent conditions are given under which a $B\in {{𝒜}}_3$ is in ${{𝒜}}_1$ and a $B\in {{𝒜}}_2$ is in ${{𝒜}}_1$.

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: ‘Classification of singularities of curves can be interpreted in dual terms as a description of “co-artin” subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)’. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer).

Let $K$ be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras ${𝒜}$ of the polynomial algebra $K[x]$ that contains the ideal $x^{m}K[x]$ for some $m\ge 1$. It is proven that the set ${𝒜}={\coprod }_{m,\mathrm{\Gamma }}{𝒜}(m,\mathrm{\Gamma })$ is a disjoint union of affine algebraic varieties (where $\mathrm{\Gamma }\coprod \{0,m,m+1,\dots \}$ is the semigroup of the singularity and $m-1$ is the Frobenius number). It is proven that each set ${𝒜}(m,\mathrm{\Gamma })$ is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on ${𝒜}(m,\mathrm{\Gamma })$. An isomorphism criterion is given for the algebras in ${𝒜}$. For each algebra $A\in {𝒜}(m,\mathrm{\Gamma })$, explicit sets of generators and defining relations are given and the automorphism group ${\text{Aut}}_K(A)$ is explicitly described. The automorphism group of the algebra $A$ is finite if and only if the algebra $A$ is not isomorphic to a monomial algebra, and in this case $|{\text{Aut}}_K(A)|<{dim}_K(A/{𝔠}_A)$ where ${𝔠}_A$ is the conductor of $A$. The set of orders of the automorphism groups of the algebras in ${𝒜}(m,\mathrm{\Gamma })$ is explicitly described.

The concept of a skew root of a skew polynomial is used to introduce some notions of algebraic closedness for $\sigma$-fields. By a $\sigma$-field, we mean a field equipped with an endomorphism. It is shown that every $\sigma$-field can be embedded in algebraically closed $\sigma$-fields of different types. Moreover, a twisted version of the method of partial fraction decomposition is introduced and studied over algebraically closed $\sigma$-fields.

Assume that ${𝒰}$ is a unital algebra over a commutative unital ring ${ℛ}$ and ${𝒮}$ is an ${𝒰}$-bimodule. A trivial extension algebra ${𝒰}⋉{𝒮}$ is defined as an ${ℛ}$-algebra with usual operations of ${ℛ}$-module ${𝒰}\times {𝒮}$ and the multiplication defined by $(u_1,s_1)(u_2,s_2)=(u_1{u}_2,u_1{s}_2+s_1{u}_2)$ for all $u_1,u_2\in {𝒰},$ $s_1,s_2\in {𝒮}.$ In this paper, we prove that under certain conditions every Jordan $n$-derivation $\mathrm{\Delta }$ on ${𝒰}⋉{𝒮}$ can be expressed as $\mathrm{\Delta }=d+\delta ,$ where $d$ is a derivation and $\delta$ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan $n$-derivations on triangular algebras and generalized matrix algebras.

We show that, making use of multiplicative lattices and idempotent endomorphisms of an algebraic structure $A$, it is possible to derive several notions concerning $A$ in a natural way. The multiplicative lattice necessary here is the complete lattice of congruences of $A$ with multiplication given by commutator of congruences. Our main application is to the study of some notions concerning left skew braces.

Let ${𝒜}$ be an ∗-algebra and $k\in {\mathbb{Z}}^{+}$. For $A,B\in {𝒜}$, the product defined by ${}_{\ast }[A,{\delta }_A{(B)}_{\lambda }^{\ast }{]}_k{=}_{\ast }{[A{,}_{\ast }{[A,{\delta }_A{(B)}_{\lambda }^{\ast }]}_{k-1}]}_1$, where ${}_{\ast }{[A,{\delta }_A{(B)}_{\lambda }^{\ast }]}_0={\delta }_A{(B)}_{\lambda }^{\ast }$ and ${}_{\ast }{[A,{\delta }_A{(B)}_{\lambda }^{\ast }]}_1=A{\delta }_A{(B)}_{\lambda }^{\ast }-{\delta }_A{(B)}_{\lambda }^{\ast }{A}^{\ast }$, is called $\lambda$-$k$-skew Lie product of $A$ and $B$, where ${\delta }_A{(B)}_{\lambda }^{\ast }=AB-\lambda B{A}^{\ast }$. This paper discusses some useful characteristics of $\lambda$-$k$-skew Lie products on prime ∗-algebras. Moreover, we investigate the effects of an additive map $\varphi$ on ${𝒜}$ with a $\lambda$-$k$-skew Lie product.

We find an expression for the Drazin inverse of a strongly $\pi$-regular element $a$ in the form $a^i{x}^j$ when $a^n=a^{n+k}x$ for some nonnegative integer $n$ and positive integer $k$. This extends the result by Azumaya, which is the case when $n=1$ and $k=1$, and a result by Drazin, which is the case when $n$ is arbitrary and $k=1$. We give new proofs of several results in the literature. For instance, we give an easy proof of the result that for two commuting Drazin invertible elements $a$, $b$ of $R$, then $a+b$ is Drazin invertible if and only if so is $1+a^{D}b$, where $a^D$ is the Drazin inverse of $a$. Our proof is akin to the case when $a$ and $b$ are invertible.

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual.

In this paper, we investigate polyform modules and introduce the notion of relatively polyform modules over a ring $R$. Some new characterizations and properties of polyform modules are obtained. For a retractable polyform module $M_R$, it is proved that the maximal right ring of quotients of End${}_R(M)$ is End${}_R(\tilde{E}(M))$, where $\tilde{E}(M)$ is the rational hull of $M$. Moreover, we characterize nonsingular modules by using the relatively polyform property. In addition, it is proved that the class of all polyform right $R$-modules is closed under submodules and rational extensions.

An algebra $A$ will be called fluid if for any linearly independent set $X$ consisting of units, the set $X^{-1}$ of inverses of the elements of $X$ is also linearly independent. We consider various examples of fluid algebras and study fluid algebras in some specific settings including direct sums, field extensions, quotient rings of the ring of polynomials over a field, and matrix algebras. Pertinent parameters for the study of fluidity are also introduced and studied.

In this paper, we report on a computation demonstrating that if $A$ and $B$ are nonempty subsets of a torsion-free group such that $AB$ has no unique product, then $\left|A\right|+\left|B\right|\ge 16$. Moreover, this bound is sharp, as there are examples where $\left|A\right|=\left|B\right|=8$, and in fact $A=B$ for at least two such examples. More generally, when $\left|A\right|$ is small, we find lower bounds on $\left|B\right|$. One consequence of this work is that any counterexample to Kaplansky’s zero-divisor conjecture must be quite complicated, if it exists at all. Another advance is that we give new examples of torsion-free groups that are not unique product groups.

Hypercyclic rings are those rings over which cyclic modules have cyclic injective hulls. Such rings were introduced by Caldwell in 1966 and studied by him and Osofsky in 1968. In this paper, we revisit Osofsky’s work on hypercyclic rings, highlighting some of her results and questions on the subject.

We provide three generalizations of all best known generating sets for $S_n$ and $A_n$: (1) $m$-cycles with consecutive entries; (2) $m$-cycles with fixed first $\ell$ entries; (3) two cycles, either $(1,\dots ,m),(1,\dots ,n)$ or $(1,\dots ,m),(2,\dots ,n)$. Our work on (3) fills a gap in the history of the problem, going back to a 1939 paper by S. Piccard [Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116(1) (1939) 752–767].

Let $D_q(n)$ be the quantized matrix algebra introduced by Dipper and Donkin. It is shown that some structural properties of $D_q(n)$ and their modules may be established and realized by means of Gröbner–Shirshov basis theory.

The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length $n{p}^s$ over ${𝔽}_{p^m}$, where $p$ is a prime number and $p$ does not divide $n$, the problem has been solved completely for all $n\le 6$ and partially for $n=7,8$. In this paper, we solve the problem for $n=9$ and all primes $p$ different from $3$ and $19$. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length $9p^s$ over ${𝔽}_{p^m}$. As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes.

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings $R$ are always “J-abelian” in the sense that each idempotent of $R$ maps onto a central idempotent in $R$/rad($R$). On the other hand, J-abelian rings turn out to be precisely the “strongly perspective rings” as well as the “strongly IC rings”, and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking $n$-th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings.

Almost all Abelian groups with the property that each subgroup isomorphic to a direct summand, is also a direct summand, are determined. The relationship with co-Hopfian groups is also addressed.

In this paper, we prove that the well-known semigroup property of being E-solid is actually equivalent to abelianness modulo the Jacobson radical in the case of (possibly non-unital) non-regular rings. We also give various other characterizations of such rings.

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic $p>0$. An alternative method, which succeeds, is the process of “hiking” a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht’s conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the “representability question.” Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht’s conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems.