Journal of Algebra and Its Applications

This study is concerned with the collections of prime and radical subacts of free acts over a semigroup $S.$ The collection of all prime subacts of any free act is topologized with two topologies as generalizations of the notion of the Zariski topology in rings and semigroups. Moreover, the interplay between some algebraic properties and some topological properties of the spectrum of free acts is investigated. In particular, the conditions under which the set of all prime subacts of free acts is a spectral space are studied. Also, it is shown that a certain type of Zariski space of free acts is isomorphic to a lattice of radical subacts.

The main purpose of this paper is to study Jacobi–Jordan-admissible algebras and some particular classes like pre-Jacobi–Jordan algebras. First, we provide characterizations of Jacobi–Jordan algebras by means of oxidation process. Then, we discuss Jacobi–Jordan-admissible algebras for which we obtain a surprising characterization as 3-power anti-associative algebras. Moreover, we introduce a concept of $\lambda$-anti-associative algebra, generalizing anti-associative algebras and prove that they are equivalent to Jacobi–Jordan-admissible algebras. Furthermore, we study the structure of pre-Jordan–Jacobi algebras which form a particular class of Jacobi–Jordan-admissible algebras. Moreover, an inductive description by a process of oxidation is provided for an interesting particular class of pre-Jacobi–Jordan algebras, called strong pre-Jacobi–Jordan algebras.

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like $W(n)$ in the theory of conservative algebras plays a similar role to the role of ${𝔤𝔩}_n$ in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra $W(n)$ for some $n\in \mathbb{N}.$ The present paper is part of a series of papers, which is dedicated to the study of the algebra $W(2)$ and its principal subalgebras.

Let $M$ be an $R$-module over a Noetherian ring $R$ and $𝔞$ an ideal of $R$ with $c=cd(𝔞,R)$. First, over an $𝔞$-relative Cohen–Macaulay local ring $(R,𝔪)$, we provide a characterization of the $𝔞$-relative sequentially Cohen–Macaulay modules $M$ in terms of $𝔞$-relative Cohen–Macaulayness of the $R$-modules ${Ext}_R^{c-i}(M,{\text{D}}_{𝔞})$ for all $i\ge 0$, where ${\text{D}}_{𝔞}={Hom}_R(H_{𝔞}^c(R),E(R/𝔪))$. Next, we prove that $M$ is finite $𝔞$-relative Cohen–Macaulay if and only if $H_i(L{\mathrm{\Lambda }}_{𝔞}(H_{𝔞}^{cd(𝔞,M)}(M)))=0$ for all $i\ne cd(𝔞,M)$ and $H_{cd(𝔞,M)}(L{\mathrm{\Lambda }}_{𝔞}(H_{𝔞}^{cd(𝔞,M)}(M)))\cong {\widehat{M}}^{𝔞}$. Finally, we provide another characterization of $𝔞$-relative sequentially Cohen–Macaulay modules $M$ in terms of vanishing of the local homology modules $H_j(L{\mathrm{\Lambda }}_{𝔞}(H_{𝔞}^i(M)))=0$ for all $0\le i\le cd(𝔞,M)$ and for all $j\ne i$.

Let $R$ be a prime ring with center $Z(R)$, let $L$ be a nonzero left ideal of $R$ and let $g,h$ be two generalized derivations of $R$. In this paper, we completely characterize the structure of $L$ and all possible forms of $g,h$ such that $g(x^m)x^n-x^{n}h(x^m)\in Z(R)$ for all $x\in L$, where $m,n$ are fixed positive integers. With this, several known results can be either deduced or generalized. Moreover, our result can be regarded as the one-sided ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317].

This paper primarily deals with the study of $G$-derivations associated with Lie–Yamaguti algebras (LY-algebras). Taking $G$ as an automorphism group, the concept of $G$-derivations, which is a derivation under both the bilinear and trilinear operations is defined for LY-algebras. Then some important properties of $G$-derivations are studied along with their relationship with other generalized derivations of LY-algebras.

Let $F$ be a real-closed field and $D$ a $F$-division algebra. In this paper, we prove that if $D$ is algebraic over $F$ then the graph $\mathrm{\Gamma }({\mathrm{\text{M}}}_n(D))$ is connected and its diameter is at most $4$, for any $n\ge 3$. If in addition, the division ring $D$ is noncommutative, we also have the same results for $n=2$. As a corollary, we show that the diameter of the commuting graph of the matrix algebra of degree $n\ge 2$ over a generalized quaternion algebra ${ℍ}_F(a,b)$, where $F$ is a real-closed field, two elements $a,b\in F^{\ast }$, is also at most $4$. This fact is a strong improvement of the previous result by Akbari et al. asserting that this diameter is at most $6$ in case $F$ is the field $ℝ$ of real numbers.

In this paper, we describe the algebras $A({\mathrm{\Gamma }}_{K_n})$ associated with the Hasse graph of the poset of the faces of the complete graph at n vertices $K_n$. Present the relations that define this algebra, calculate the Hilbert series of quadratic grade algebra $A({\mathrm{\Gamma }}_{K_n})$ and the graded trace generating functions of $D_n$ acting on $A({\mathrm{\Gamma }}_{K_n})$ and show that $A({\mathrm{\Gamma }}_{K_n})$ is a Koszul algebra. The methodology adopted in this paper consists of building the algebra $A({\mathrm{\Gamma }}_{K_n})$ using [I. Gelfand, V. Retakh, S. Serconek and R. L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. 11(2) (2005) 281], giving a presentation based on what was established by [V. Retakh, S. Serconek and R. L. Wilson, On a class of koszul algebras associated to directed graphs, J. Algebra 304(2) (2006) 1114–1129] in terms of the vertices of the graph ${\mathrm{\Gamma }}_{K_n}$. After that, we use [V. Retakh, S. Serconek and R. L. Wilson, Hilbert series of algebras associated to directed graphs, J. Algebra 312(1) (2007) 142–151] to determine the Hilbert series of $A({\mathrm{\Gamma }}_{K_n})$ and use [9] again to show that this algebra is a Koszul algebra. We use [J. Caldeira, A. De Souza Lima and J. Eder Salvador De Vasconcelos, Representations of automorphism groups of algebras associated to star polygons, J. Algebra Appl. 18(10) (2019) 1950197; C. Duffy, Representations of Aut(A($\mathrm{\Gamma }$)) acting on homogeneous components of $A(\mathrm{\Gamma })$ and $A(\mathrm{\Gamma })!$, Adv. Appl. Math. 42(1) (2009) 94–122] to determine the generating functions of the graduated trace of $D_n$ acting on $A({\mathrm{\Gamma }}_{K_n})$.

The authors of A. Benhissi and S. Hizem [Nonnil-Noetherian rings and SFT property, Rocky Mountain J. Math. 41 (2011) 1483–1500] proved that the primary decomposition of a nonnil-ideal of a nonnil-Noetherian ring always exists. We will show in this paper that this primary decomposition is minimal. We next prove that over any nonnil-Noetherian ring, a sub-class of modules admit a primary decomposition. In the second part of this paper, we give some properties related to $\varphi$-chained and $\varphi$-Prüfer rings.

We discuss and relate finiteness conditions for certain field invariants which are studied in quadratic form theory. This includes the $u$-invariant, the reduced stability index and the symbol lengths for Galois cohomology groups with coefficients in ${\mu }_2=\{+1,-1\}$, as well as a new invariant called the splitting height.

An additive mappings $\delta$ on $R$ is called a derivation if $\delta (ab)=\delta (a)b+a\delta (b)$ for all $a,b\in R$. If $\delta$ is a derivation on a ring $R$, $M$ and $N$ are right $R$-modules and $f$ is a right $R$-linear mapping from $M$ to $N$, then an additive mapping $d:M\to N$ is called a $(\delta ,f)$-derivation if $d(xa)=d(x)a+f(x)\delta (a)$ for all $x\in M$ and $a\in R$. If $\delta$ is determined, then the $(\delta ,f)$-derivation is written briefly as $f$-derivation. The aim of this paper is to initiate the study of $f$-derivations on polynomial modules. In fact, we induce the derivations of polynomial rings $R[x]$ based on the derivations of rings. In addition, we define $R[x]$-linear mapping on a polynomial module over a polynomial ring based on $R$-linear mapping on modules. Finally, we induce the $f$-derivations on polynomial modules over a polynomial ring based on $f$-derivations of modules and apply this result to the semigroup rings.

For a semisimple quasi-triangular Hopf algebra $(H,R)$ over a field $k$ of characteristic zero, and a strongly separable quantum commutative $H$-module algebra $A$, we show that $A\#H$ is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra $End{A}^{\ast }\otimes H$. With these structures, ${}_{A\#H}Mod$ is the monoidal category introduced by Cohen and Westreich, and ${}_{End{A}^{\ast }\otimes H}{ℳ}$ is tensor equivalent to ${}_H{ℳ}$. If $A$ is in the Müger center of ${}_H{ℳ}$, then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.

Rota–Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota–Baxter operators on Leibniz algebras. We investigate modified Rota–Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota–Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.

Let $G$ be a finite group and $\mathrm{\text{Irr}}(G)$ be the set of irreducible characters of $G$. The codegree of an irreducible character $\chi$ of the group $G$ is defined as $\mathrm{\text{cod}}(\chi )=|G:\mathrm{\text{ker}}(\chi )|/\chi (1)$. In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the $p$-parts of the codegrees and character degrees.

In the paper, we discuss the partial actions of Sweedler Hopf algebra on the generalized quaternion algebra, and give the sufficient and necessary conditions to make the actions be the partial actions. As an application, we give a complete description of all partial actions of Sweedler Hopf algebra on the generalized quaternion algebra.

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring $R$ is reduced FPF if and only if it is $\mathrm{\Pi }$-semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a $C(X)$ to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space $X$ is fraction-dense if and only if $Q_{\mathrm{\text{cl}}}(X)$ is a continuous ring.

In [Z. Fan, H. Ma and H. Xiao, Equivariant K-theory approach to $ı$-quantum groups, Publ. Res. Inst. Math. Sci. 58 (2022), no.3, 635–668], it is proved that the convolution algebra of the top Borel–Moore homology on the Steinberg variety of type $B/C$ realizes $U(s{l}_n^{𝜃})$, where $s{l}_n^{𝜃}$ is the fixed point subalgebra of involution on $s{l}_n$. So, the top Borel–Moore homology of the partial Springer fibers gives the representations of $U(s{l}_n^{𝜃})$. In this paper, we study these representations using Schur–Weyl duality and Springer theory.

In this paper, we propose the concept of an $\mathrm{\Omega }$-Rota–Baxter system, which is a generalization of a Rota–Baxter system and an $\mathrm{\Omega }$-Rota–Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis of a free $\mathrm{\Omega }$-Rota–Baxter system for an extended diassociative semigroup $\mathrm{\Omega }$, in terms of bracketed words and the method of Gröbner–Shirshov bases. As applications, we introduce the concepts of Rota–Baxter system family algebras and matching Rota–Baxter systems as special cases of $\mathrm{\Omega }$-Rota–Baxter systems, and construct their free objects. Meanwhile, free $\mathrm{\Omega }$-Rota–Baxter algebras of weight zero, free Rota–Baxter systems, free Rota–Baxter family algebras and free matching Rota–Baxter algebras are reconstructed via new method.

A subset $S$ of an integral domain is called a semidomain if the pairs $(S,+)$ and $(S\backslash \{0\},\cdot )$ are commutative and cancellative semigroups with identities. The multiplication of $S$ extends to the group of differences $\mathcal{𝒢}(S)$, turning $\mathcal{𝒢}(S)$ into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e. semidomains $S$ for which either $s\in S$ or $-s\in S$ for every $s\in \mathcal{𝒢}(S)$). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the paper, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.

Let ${𝒜}$ be a unital prime ∗-algebra containing a nontrivial projection $P_1$. In this paper, it is shown that if a mapping $\mathrm{\Phi }:{𝒜}\to {𝒜}$ satisfies

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, we prove that if every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal, then $G$ is $p$-nilpotent or $p$-closed for each prime divisor $p$ of $\left|G\right|$, which improves a theorem obtained by Beltr$́a$n and Shao.