Journal of Algebra and Its Applications

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let $G$ be a unicyclic graph with a unique odd cycle and $I=I(G)$ be its edge ideal. We compute the exact values of all symbolic defects of $I$ using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal $I$. We classify the class of unicyclic graphs when some power of maximal ideal annihilates $I^{(s)}/I^s$ for any fixed $s$. Also for those class of graphs, we compute the Hilbert function of the module $I^{(s)}/I^s$ for all $s.$

Let $A$ be an *-algebra with identity $1_A$ and $e_1$ and $e_2=1_A-e_1$ nontrivial symmetric idempotents in $A$. In this paper we study the characterization of nonlinear mixed *-Jordan-type derivations. In particular, if ${ℳ}$ is a factor von Neumann algebra then every unital nonlinear mixed *-Jordan-type derivations are additive *-derivations.

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma (G)=\{{\sigma }_i|{\sigma }_i\cap \pi (G)\ne \varnothing \}$. A set ${\mathscr{H}}$ of subgroups of $G$ is said to be a complete Hall$\sigma$-set of $G$ if every non-identity member of ${\mathscr{H}}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i\in \sigma$ and ${\mathscr{H}}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma (G)$. A subgroup $H$ of $G$ is said to be $\sigma$-primary if it is a finite ${\sigma }_i$-group for some $i$; $\sigma$-abnormal in $G$ if $K/L_K$ is not $\sigma$-primary whenever $H\le L<K\le G$ and $L$ is a maximal subgroup of $K$. A subgroup $H$ of $G$ is called $𝔉$-subnormal if there is a chain of subgroups $H=H_0<\cdot H_1<\cdot \dots <\cdot H_t=G$ such that $H_i/{(H_{i-1})}_{H_i}\in 𝔉$ for all $i$. This is equivalent to $H_i^{𝔉}\le H_{i-1}$. In this paper, we study the structure of a finite group $G$ in which every $\sigma$-primary cyclic subgroups are $\sigma$-quasinormal or $\sigma$-abnormal. We also describe the structure of a finite group $G$ with self-normalizing or $𝔉$-subnormal $\sigma$-primary cyclic subgroups for a subgroup-closed saturated lattice formation $𝔉$ containing all $\sigma$-nilpotent groups.

A unit-picker is a map ${𝒢}$ that associates to every ring $R$ a well-defined set ${𝒢}(R)$ of central units in $R$ which contains $1_R$, is invariant under isomorphisms of rings, is closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let ${𝒢}$ be a unit-picker. A ring $R$ is called $U_{{𝒢}}U$ if $U(R)={𝒢}(R)+\mathrm{\text{nil}}(R)$ and $U_{{𝒢}}J$ if $U(R)={𝒢}(R)+J(R)$ and $U_{{𝒢}}NJ$ if $U(R)={𝒢}(R)+\mathrm{\text{nil}}(R)+J(R)$. An extensive study of these rings is conducted, and their connections with strongly nil ${𝒢}$-clean rings and semi ${𝒢}$-Boolean rings are investigated. When ${𝒢}$ is specified, known results of $UU$-rings, $UJ$-rings and $UNJ$-rings are obtained and new results are proved.

Recall that a commutative ring $R$ is a locally integral domain if its localization $R_P$ is an integral domain for each prime ideal $P$ of $R.$ Our aim in this paper is to extend the notion of locally integral domains to modules. Let $R$ be a commutative ring with a unity and $M$ a nonzero unital $R$-module. $M$ is called a locally torsion-free module if the localization $M_P$ of $M$ is a torsion-free $R_P$-module for each prime ideal $P$ of $R.$ In addition to giving many properties of locally torsion-free modules, we use them to characterize Baer modules, torsion free modules, and von Neumann regular rings.

Describing the group of units of a group ring is a classical problem. Let $p$ be a rational prime number. We set ${𝜃}_p$ a primitive root of unity of order $p$, $\mathbb{Z}[{𝜃}_p]$ the ring of $p$-cyclotomic integers, $G$ a finite abelian $p$-group and $U_1(\mathbb{Z}[{𝜃}_p]G)$ the group of the units $u$ of $\mathbb{Z}[{𝜃}_p]G$ such that $𝜀(u)\equiv 1\mathrm{\text{mod}}({𝜃}_p-1)$, where $𝜀$ is the augmentation map. We will prove that all the elements of the group $U_1(\mathbb{Z}[{𝜃}_p]G)$ arise from the units of the group ring $\mathbb{Z}(C_p\times G)$, where $C_p$ is the cyclic group of order $p$. As an application, we describe explicitly the group of units of the group ring $\mathbb{Z}[{𝜃}_p]G$ when $G$ is an elementary abelian $p$-group and $p<67$ is a regular prime number.

Let * be an involution of a non-commutative prime ring $\Re$ with the maximal symmetric ring of quotients and the extended centroid of $\Re$ denoted by $Q_{ms}(\Re )$ and $C$, respectively. Consider ${𝒢}:\Re \to Q_{ms}(\Re )$ be an additive map, if ${𝒢}(u^2)-{𝒢}(u)u^{*}-u{𝒢}(u)\in C$ for all $u\in \Re$, then such a map ${𝒢}$ is termed as a weak Jordan *-derivation. With the smart handling of the FI-theory and facing the challenging case of low dimensions, we prove that every weak Jordan *-derivation of $\Re$ is $X$-inner unless ${\text{dim}}_C\Re C\le 9$. Moreover, if * is of the first kind, then every weak Jordan *-derivation ${𝒢}$ of $\Re$ is $X$-inner if and only if ${𝒢}$ is $Z(\Re )$-linear.

We continue the study of the reflexivity of ideals, introduced by Mason, and extend this notion to the subsets in rings. We first construct the smallest reflexive ideal containing $S$ from any proper ideal $S$ of any given ring $R$; by which we can construct reflexive ideals but not semiprime in a kind of noncommutative ring. A subset $T$ of a ring $R$ is called reflexively closed if $aRb\subseteq T$ for $a,b\in R$ implies $bRa\subseteq T$, checking that a ring $R$ is symmetric if and only if the right (left) annihilator of $a$ is reflexively closed for any $a\in R$. We prove that the set of all nilpotent elements in a ring $R$ is reflexively closed if and only if $aR$ is nil for any nilpotent element $a$ in $R$; and that the Köthe’s conjecture holds if and only if the union (sum) of the upper nilradical and any nil right ideal is reflexively closed. We provide another process to show that the set of all nilpotent elements of the polynomial ring over an NI ring need not be reflexively closed.

This paper details the diassociative analogue of results concerning the Schur multiplier and other extension-theoretic concepts that originate in group theory. We first prove that covers of diassociative algebras are unique. Second, we show that the multiplier of a diassociative algebra is characterized by the second cohomology group with coefficients in the field. Third, we establish criteria for when the center of a cover maps onto the center of the algebra. Along the way, we obtain a collection of exact sequences, characterizations, and a brief theory of unicentral diassociative algebras and stem extensions. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras.

An Einstein nilradical is a nilpotent Lie algebra which can be the nilradical of an Einstein metric solvable Lie algebra. A Lie algebra is called complete if its center is zero and all its derivations are inner. A nilpotent Lie algebra is called completable if it is the maximal nilpotent ideal of a complete solvable Lie algebra. In this short note, based on the classification result of Einstein metric solvable Lie algebras, we show that any Einstein nilradical is completable. This provides a purely algebraic obstruction for a nilpotent Lie algebra to be an Einstein nilradical.

Let $K=\mathbb{Q}(𝜃)$ be an algebraic number field with $𝜃$ a root of an irreducible trinomial $f(x)=x^5+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, we compute the highest power of each prime $p$ dividing the discriminant of $K$ in terms of powers of $p$ dividing $a,b$ and the discriminant of $f(x)$ besides explicitly constructing a $p$-integral basis of $K$. These $p$-integral bases lead to the construction of an integral basis of $K$ which is illustrated with examples.

Let $R=k[x_1,\dots ,x_n]$ be the polynomial ring over a field $k$, $G$ be a graph on vertex set $V=\{x_1,\dots ,x_n\}$ and $I=I(G)$ be its edge ideal. In this paper, we study bi-sequentially Cohen–Macaulay (bi-SCM) bipartite graphs and as consequence we classify all bi-SCM tree graphs. Furthermore, if $R/I$ is bi-SCM then we determine the projective dimension of $R/I$. Moreover, we give some examples.

Let $R$ be a ring with finite Gorenstein weak global dimension. We characterize Gorenstein projective, injective and flat modules over $R$. As an application, it is proved that $R$ is (strongly) CM-free if and only if $R$ has finite weak global dimension.

In this paper, we introduce the concept of rings with right (left) weak property (A) which is a generalization of rings with right (left) property (A). A ring $R$ has right (left) weak property (A) if every finitely generated two-sided ideal $I$ of $R$ with $I\subseteq Z_l(R)(Z_r(R))$, there exists nonzero $u\in R$ $(v\in R)$ such that $u\in Nr\cdot {\mathrm{\text{ann}}}_R(I)$ $(v\in Nl\cdot {\mathrm{\text{ann}}}_R(I))$. Further, we study various extensions of rings with weak property (A) including matrix rings, polynomial rings and Ore extensions.

There has been a great deal of interest lately in studying rings in which every element can be written as a sum of potent elements and nilpotent elements. In this paper, we focus primarily on rings in which every element can be written as the sum of some number of $m$-potent elements and a nilpotent (for some choice of $m$), all of which commute with each other. The research that has been conducted thus far in this area has revealed a number of interesting results and intriguing patterns. Our goal in this paper is to unify and extend these results, for the purpose of organizing the current knowledge and facilitating future research in this area.

A right module $M$ is extending if every submodule is essential in a direct summand of $M$. In this paper, necessary and sufficient conditions are obtained for a trivial Morita context ring to be right extending.

Let $R$ be a prime ring, let $L$ be a noncentral Lie ideal of $R$ and let $g,h$ be two generalized derivations of $R$. In this paper, we characterize the structure of $R$ and all possible forms of $g$ and $h$ such that ${[g(x^m)x^n-x^{s}h(x^t),x^r]}_k=0$ for all $x\in L$, where $m,n,s,t,r,k$ are fixed positive integers. With this, several known results can be either deduced or generalized. In particular, we give a Lie ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317] and describe a more complete version of the theorem recently obtained by Dhara and De Filippis in [Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020) 154–167].

In a paper from 2002, Zhang [Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131–4151] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. Following his approach, we show that for amalgamated duplication of dual Banach algebras, we are able to choose appropriate conditions for weak Connes amenability of those Banach algebras which extend the Zhang’s result in a much more general setting. Moreover, apart from the characterization of minimal idempotents, we restrict our investigation to the study of injectivity, projectivity and flatness of amalgamated duplication of Banach algebras. Particularly, it is worth mentioning that our result paves the way for studying of homological properties and weak Connes amenability of those dual Banach algebras that have a semidirect product-like structure including Lau algebras.

We prove that any Novikov algebra over a field of characteristic $\ne 2$ is Lie-solvable if and only if its commutator ideal $[N,N]$ is right nilpotent. We also construct examples of infinite-dimensional Lie-solvable Novikov algebras $N$ with non-nilpotent commutator ideal $[N,N]$.

Given a finite group $G$, we study certain regular subgroups of the group of permutations of $G$, which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to $G$ and Hopf–Galois structures admitted by a Galois extension of fields with Galois group isomorphic to $G$. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf–Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf–Galois structure is determined by the corresponding skew left brace. We investigate these questions in the context of a variety of existing constructions in the literature. As an application of our results we classify the isomorphically distinct Hopf algebras that give Hopf–Galois structures on a Galois extension of degree $pq$ for $p>q$ prime numbers.

The concepts of derivation and centroid for Lie–Yamaguti algebras are generalized in this paper. A quasi-derivation of a LY-algebra can be embedded as a derivation in a larger LY-algebra. The relationship between quasi-derivations and robustness of Lie–Yamaguti algebras has been studied.

Let $T=\left(\right.\begin{array}{cc}\hfill R\hfill & \hfill V\hfill \\ \hfill W\hfill & \hfill S\hfill \end{array}\left)\right.$ be a ring of the Morita context. We define a concept of Dubrovin valuation modules for bi-modules $V$ and $W$ and give necessary and sufficient conditions for $T$ to be a Dubrovin valuation ring in terms of $R$, $S$, $V$ and $W$.

We review various constructions of wreath products of groups, semigroups, Lie algebras and associative algebras and discuss their realizations in matrix wreath products of associative algebras. As an application we prove a new version of Evans’s embedding theorem [T. Evans, Embedding theorems for multiplicative systems and projective geometries, Proc. American Math. Soc. 3 (1952) 614–620].