Journal of Algebra and Its Applications

In this paper, we study the equivalence of quantum stabilizer codes via symplectic isometries of stabilizer codes. We define monomially and symplectically equivalent stabilizer codes and determine how different the two notions can be. Further, we show that monomial maps correspond to local Clifford operators. We relate the latter with the LU-LC Conjecture.

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper, we establish such a generalization of the “salamander lemma” due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Márki and W. Tholen and exact categories in the sense of M. Grandis.

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras.

In this work, we show that the existence of invertible generalized derivations impose strong restrictions on the structure of a complex finite-dimensional Lie algebra. In particular, we recover the fact that a real Lie algebra admitting an abelian complex structure is necessarily solvable. On the other hand, we state a structure theorem for a Lie algebra $𝔤$ admitting a periodic generalized derivation $D\in {Der}_{(-1,1,1)}(𝔤)$.

Let $R$ be a prime ring with characteristic different from two, $d$ a derivation of $R$, $L$ a noncentral Lie ideal of $R$, and $a\in R$. In the present paper, it is shown that if one of the following conditions holds: (i) $a{(d{(uv)}^s-{(uv)}^t)}^n=0$, (ii) $a{(d{(uv)}^s+{(uv)}^t)}^n=0$, (iii) $a{(d{(uv)}^s-{(vu)}^t)}^n=0$ and (iv) $a{(d{(uv)}^s+{(vu)}^t)}^n=0$ for all $u,v\in L$, where $n,s,t$ are fixed positive integers, then $a=0$ unless $R$ satisfies $s_4$, the standard polynomial identity in four variables.

Let $R$ be a commutative Noetherian domain, $M$ a nonzero $R$-module of finite injective dimension, and $I$ be a nonzero ideal of $R$. In this paper, we prove that whenever $\mathrm{\text{injdim}}M=cd(I,M)=t$, then the annihilator of $H_I^t(M)$ is zero. Also, we calculate the annihilator of $H_I^{n+t}(N,M)$ for finitely generated $R$-modules $N$ and $M$ with conditions $\mathrm{\text{projdim}}N=n<\infty$ and $\mathrm{\text{injdim}}M=t<\infty$. Moreover, if $(R,𝔪)$ is a regular Noetherian local ring and $𝔭\in \mathrm{\text{Spec}}(R)$ such that $d=\mathrm{\text{dim}}\frac{R}{𝔭}\ge 2$, then we show that there exists an ideal $J$ of $R$ such that $𝔭\subseteq J$, $ht\frac{J}{𝔭}=1$ and $J^n{H}_{𝔪}^1(\frac{R}{{𝔭}^n})=0$.

We study the Betti cone of modules of a standard graded algebra over a field, and find a class of modules whose Betti diagrams span extremal rays of the Betti cone. We also identify a class of one-dimensional Cohen–Macaulay rings with certain Hilbert series, for which the Betti cone is spanned by these extremal rays. These results lead to an algorithm for the decomposition of Betti table of modules, into the extremal rays, over such rings, and also help to obtain bounds for the multiplicity of the given module.

Let ${{ℱ}}_{\mu }$ be the space of $\mu$-densities over $S^1$, we compute the space ${\mathrm{\text{H}}}_{\mathrm{\text{diff}}}^1(𝔞𝔣𝔣(1);{{𝒟}}_{\underline{\lambda },\mu })$ where $\underline{\lambda }=({\lambda }_1,\dots ,{\lambda }_n)\in {ℂ}^n$ and ${{𝒟}}_{\underline{\lambda },\mu }$ is the space of $n$-ary linear differential operators from ${{ℱ}}_{{\lambda }_1}\otimes \cdots \otimes {{ℱ}}_{{\lambda }_n}$ to ${{ℱ}}_{\mu }$. We also compute the simplest super analog spaces ${\mathrm{\text{H}}}_{\mathrm{\text{diff}}}^1(𝔞𝔣𝔣(1|1);{𝔇}_{\underline{\lambda },\mu })$.

An element $a$ in a ring $R$ has a gs-Drazin inverse if there exists $p^2=p\in {\mathrm{\text{comm}}}^2(a)$ such that $a-p\in R^{\mathrm{\text{qnil}}}$. In this paper, we extend Cline’s formula and Jacobson’s Lemma for gs-Drazin inverses. Various additive properties of gs-Drazin inverses are thereby obtained.

The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra ${𝕀}_n:=K〈x_1,\dots ,x_n,\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n},{\int }_1,\dots ,{\int }_n〉$ of polynomial integro-differential operators and the Jacobian algebra ${𝔸}_n$ is equal to $n$ (over a field of characteristic zero). The algebras ${𝕀}_n$ and ${𝔸}_n$ are neither left nor right Noetherian and ${𝕀}_n\subset {𝔸}_n$. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert’s Syzygy Theorem is proven for the algebras ${𝕀}_n$, ${𝔸}_n$ and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras ${𝕀}_n$ and ${𝔸}_n$ is $n$ and the weak global dimension of all the factor algebras of ${𝕀}_n$ and ${𝔸}_n$ is $n$.

The concept of isoclinism on the class of groups was introduced by Philip Hall in 1940. Then, in 1993, Moneyhun used this concept on Lie algebras. In this paper, we study the notion of isoclinism on pairs of Lie rings, and some results are obtained when two isoclinic pairs of Lie rings are given. In particular, we prove that every isoclinism class of pairs of Lie rings contains at least one stem pair.

Let $(R,𝔪)$ be a commutative Noetherian complete local ring and $𝔞$ and $𝔟$ ideals of $R$. Motivated by a question of Rees, we study the relationship between $\overline{𝔟}$, the classical Northcott–Rees integral closure of $𝔟$, and ${𝔟}^{\ast (H)}$, the integral closure of $𝔟$ relative to an Artinian $R$-module $H$ (also called here ST-closure of $𝔟$ on $H$), in order to study a relation between $e(𝔞)$, the multiplicity of $𝔞$, and $e^{\prime }(𝔞;H)$, the multiplicity of $𝔞$ relative to an Artinian $R$-module $H$. We conclude $\overline{𝔟}={𝔟}^{\ast (H)}$ when every minimal prime ideal of $R$ belongs to the set of attached primes of $H$. As an application, we show what happens when $H$ is a generalized local cohomology module.

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if $𝔞$ is a nonzero proper ideal of a subtractive valuation semiring $S$ then $𝔞$ is a 2-absorbing ideal of $S$ if and only if $𝔞=𝔭$ or $𝔞={𝔭}^2$ where $𝔭=\sqrt{𝔞}$ is a prime ideal of $S$. We also show that each 2-absorbing ideal of a subtractive semiring $S$ is prime if and only if the prime ideals of $S$ are comparable and if $𝔭$ is a minimal prime over a 2-absorbing ideal $𝔞$, then $𝔞𝔪=𝔭$, where $𝔪$ is the unique maximal ideal of $S$.

A formula for the number of gradations, up to equivalence, of cyclic rings by cancellative monoids is given. As an application, the nil and Jacobson radicals of cyclic rings are shown to be homogeneous.

In this paper, we prove that the direct product $\mathrm{\text{Sz}}(q_1)\times \mathrm{\text{Sz}}(q_2)\times \cdots \times \mathrm{\text{Sz}}(q_k)$, where $q_i=2^{2n_i+1}\ge 8$ are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product $\mathrm{\text{Sz}}(2^{p_1})\times \mathrm{\text{Sz}}(2^{p_2})\times \cdots \times \mathrm{\text{Sz}}(2^{p_k})$, where $p_i$’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

In this paper, we determine the behavior of the $c$-nilpotent multiplier of Lie algebras with respect to the direct sum. Then we give some results on the $c$-capability of the direct sum of finite dimensional Lie algebras.

The set of all maximal ideals of the ring ${ℳ}(X,{𝒜})$ of real valued measurable functions on a measurable space $(X,{𝒜})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $\widehat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of ${ℳ}(X,{𝒜})$ in terms of the points of $\widehat{X}$. It is further shown that the structure spaces of all the intermediate subrings of ${ℳ}(X,{𝒜})$ containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when $X$ is a $P$-space, then $C(X)={ℳ}(X,{𝒜})$, where ${𝒜}$ is the $\sigma$-algebra consisting of the zero-sets of $X$.

If $R$ is a finite commutative ring, it is well known that there exists a nonzero polynomial in $R[T]$ which is satisfied by every element of $R$. In this paper, we classify all commutative rings $R$ such that every element of $R$ satisfies a particular monic polynomial. If the polynomial, satisfied by the elements of $R$, is not required to be monic, then we can give a classification only for Noetherian rings, giving examples to show that the characterization does not extend to arbitrary commutative rings.

The power graph ${𝒫}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity $\kappa ({𝒫}(G))$ of ${𝒫}(G)$ is the minimum number of vertices which need to be removed from $G$ so that the induced subgraph of ${𝒫}(G)$ on the remaining vertices is disconnected or has only one vertex. For a positive integer $n$, let $C_n$ be the cyclic group of order $n$. Suppose that the prime power decomposition of $n$ is given by $n=p_1^{n_1}{p}_2^{n_2}\cdots p_r^{n_r}$, where $r\ge 1$, $n_1,n_2,\dots ,n_r$ are positive integers and $p_1,p_2,\dots ,p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. The vertex connectivity $\kappa ({𝒫}(C_n))$ of ${𝒫}(C_n)$ is known for $r\le 3$, see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for $r\ge 4$, we give a new upper bound for $\kappa ({𝒫}(C_n))$ and determine $\kappa ({𝒫}(C_n))$ when $n_r\ge 2$. We also determine $\kappa ({𝒫}(C_n))$ when $n$ is a product of distinct prime numbers.

We retract [1, Lemma 3.4] as the statement is incorrect. In consequence, we correct the statements of Theorems 1.2 and 4.5 and their proofs.