Journal of Algebra and Its Applications

Let $(R,𝔪)$ be a regular local ring of dimension $d\ge 2$. A local monoidal transform of $R$ is a ring of the form $R_1=R{[\frac{𝔭}{x}]}_{{𝔪}_1}$, where $x\in 𝔭$ is a regular parameter, $𝔭$ is a regular prime ideal of $R$ and ${𝔪}_1$ is a maximal ideal of $R[\frac{𝔭}{x}]$ lying over $𝔪.$ In this paper, we study some features of the rings $S={\cup }_{n\ge 0}^{\infty }{R}_n$ obtained as infinite directed union of iterated local monoidal transforms of $R$. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

The enhanced power graph ${{𝒢}}_e(G)$ of a group $G$ is the graph with vertex set $G$ such that two vertices $x$ and $y$ are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups $G$ for which $\mathrm{\text{Aut}}({{𝒢}}_e(G)$ is abelian, all finite groups $G$ with $|\mathrm{\text{Aut}}({{𝒢}}_e(G)|$ being prime power, and all finite groups $G$ with $|\mathrm{\text{Aut}}({{𝒢}}_e(G)|$ being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

In this paper, we study the structure of finite type epimorphisms of rings.

We study complex finite-dimensional Leibniz algebra bimodule over ${𝔰𝔩}_2$ that as a Lie algebra module is split into a direct sum of two simple ${𝔰𝔩}_2$-modules. We prove that in this case there are only two nonsplit Leibniz ${𝔰𝔩}_2$-bimodules and we describe the actions.

For $n\ge 2$ and for a ring $R$, the notation $P_n(R)$ means that $a^n-a$ is nilpotent for all $a\in R$, and $Q_n(R)$ means that $R/J(R)$ has identity $x^n=x$ and $J(R)$ is nil, where $J(R)$ is the Jacobson radical of $R$. In this paper, rings $R$ for which $P_n(R)$ holds are completely characterized for integers $n$ up to $12$, for $n=2^k$ with $k$ odd, and for $n=2^{k}p$, where $k>0$ and $p=3,5,7$ or $9$. For an arbitrary integer $n\ge 2$, we prove that $P_n(R)\iff Q_n(R)$ for all rings $R$ if and only if $n$ is even with $n\equiv 1(mod3)$, or $n$ is odd with $n\not\equiv 1(mod3)$ and $n\not\equiv 1(mod8)$.

In this paper, we investigate Gorenstein FC-projective modules and Gorenstein FC-projective dimensions, and characterize rings over which every module is Gorenstein FC-projective and rings over which every submodule of a projective is Gorenstein FC-projective, respectively. Under an almost excellent extension of rings, we obtain several invariant properties of Gorenstein FC-projective modules and Gorenstein FC-projective dimensions.

In this paper, we study polynomials of the form $f(x)=x^{\frac{q^n-1}{q-1}+1}+bx\in {𝔽}_{q^n}[x]$, where $n=5$ and list all permutation polynomials (PPs) and complete permutation polynomials (CPPs) of this form. This type of polynomials were studied by Bassalygo and Zinoviev for the cases $n=2$ and $n=3$, Wu, Li, Helleseth and Zhang for the case $n=4$, $p\ne 2$, Bassalygo and Zinoviev answered the question for the case $n=4$, $p=2$ and finally by Bartoli et al. for the case $n=6$. Here, we determine all PPs and CPPs for the case $n=5$.

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers $n$ and $\mathrm{\Delta }$ with $3\le \mathrm{\Delta }\le n-2$, we prove that among the connected graphs on $n$ vertices of given maximum degree $\mathrm{\Delta }$ with at least one cycle, the graph $U(n,\mathrm{\Delta })$ uniquely maximizes the distance spectral radius, where $U(n,\mathrm{\Delta })$ is the graph obtained from the disjoint star on $\mathrm{\Delta }$ vertices and path on $n-\mathrm{\Delta }$ vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

We introduce the notion of Fitting characters for arbitrary finite groups, and prove that under some conditions the product of these characters is irreducible and the unique factorization of this form also holds. Moreover, we show that any nonlinear quasi-primitive character of solvable groups can be uniquely factored (up to multiplication by linear characters) as the product of certain Fitting characters on some extension groups.

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form $L={𝒰}+({\sum }_j{I}_j)$ with ${𝒰}$ a linear subspace of $H$ and any $I_j$ a well-described (split) ideal of $L$ satisfying $[I_{j,\bar{0}},I_k]=\{I_{j,ī},I_{k,ī},L_{ī}\}=0$, with $ī\in \{\bar{1},\bar{2}\}$, if $j\ne k$. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that $L$ is the direct sum of the family of its (split) simple ideals) is stated.

We use two main techniques, namely, residuation and separators of points, to show that the Hilbert function of a certain fat point set supported on a grid complete intersection is the same as the Hilbert function of a reduced set of points called a partial intersection. As an application, we answer a question of Tohǎneanu and Van Tuyl which relates the minimum Hamming distance of a special linear code and the minimum socle degree of the associated fat point set.

For any automorphism of a crystallographic group of ${\mathrm{\text{Sol}}}_1^4$, we prove that its Reidemeister number is infinite.

We give a formula for the subgroup normality degree $P_N(H,G)$ of a subgroup $H$ in a finite group $G$, and determine subgroup normality degrees in the case where $G$ is a finite $p$-group of order $p^5$ or a finite $p$-group with a cyclic subgroup of index $p^2$.

In this paper, we study some graph-theoretic properties about the zero-divisor graph $\mathrm{\Gamma }(Q)$ of a finite quasi-ordered set $Q$ with a least element 0 and its line graph $L(\mathrm{\Gamma }(Q))$. First, we offer a method to find all the minimal prime ideals of a quasi-ordered set. Especially, this method is applicable for a partially ordered set. Then, we completely characterize the diameter and girth of $\mathrm{\Gamma }(Q)$ by the minimal prime ideals of $Q$. Besides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete $k$-partite graphs. We also investigate the planarity of $\mathrm{\Gamma }(Q)$. Finally, we obtain the characterization for the line graph $L(\mathrm{\Gamma }(Q))$ in terms of its diameter, girth and planarity.

For a right module $M_R$ over a ring $R$, we consider the set $I$ of all the endomorphisms $\phi \in E:=\mathrm{\text{End}}(M_R)$ that are not injective and the set $K$ of all the endomorphisms $\phi$ that are not surjective. We prove that when $M_R$ is a uniserial module, then $E/K$ is a left chain domain and $E/I$ is a right chain domain. The technique we make use of to prove this can be applied to arbitrary modules $M_R$, not-necessarily uniserial. When the endomorphisms $\phi$ are not in $I$ (not in $K$), then left (right) divisibility in $E$ corresponds to inclusion in the lattice ${ℒ}(M_R)$ of all submodules of $M_R$. This allows to study factorizations of injective (surjective, respectively) endomorphisms of $M_R$ analyzing finite chains in the partially ordered set ${ℒ}(M_R)$.

We study rings in which $ab$ nonzero idempotent implies $ba$ is also an idempotent. We call such rings i-reversible. Besides studying the basic properties of i-reversible rings, we characterize i-reversible triangular matrix rings, i-reversible matrix rings over commutative rings and i-reversible exchange rings.

The solubility of a finite group with less than $53$ supersoluble subgroups is confirmed in this paper. Moreover, we prove that a finite insoluble group has exactly $53$ supersoluble subgroups if and only if it is isomorphic to $A_5$. Furthermore, it is shown that a finite group with less than $7$ conjugacy classes of supersoluble subgroups is soluble, and the only finite insoluble group with $7$ conjugacy classes of supersoluble subgroups is isomorphic to $A_5$.

In this paper, we introduce the concept of $p$-semisimple modules. We prove that a multiplication reduced module is $p$-semisimple if and only if it is a Baer module. We show that a large family of abelian groups are $p$-semisimple. Furthermore, we give a topological characterizations of type submodules (ideals) of multiplication reduced modules ($J$-semisimple rings). Moreover, we observe that there is a one-to-one correspondence between type ideals of some algebraic structures on one hand and regular closed subsets of some related topological spaces on the other hand. This also characterizes the form of closed ideals in $C(X)$.

Let $G$ be a unique product group, i.e. for any two finite subsets $A,B$ of $G$, there exists $x\in G$ which can be uniquely expressed as a product of an element of $A$ and an element of $B$. We prove that if $C$ is a finite subset of $G$ containing the identity element such that $〈C〉$ is not abelian, then, for all subsets $B$ of $G$ with $\left|B\right|\ge 7$, $\left|BC\right|\ge \left|B\right|+\left|C\right|+2$. Also, we prove that if $C$ is a finite subset containing the identity element of a torsion-free group $G$ such that $\left|C\right|=3$ and $〈C〉$ is not abelian, then for all subsets $B$ of $G$ with $\left|B\right|\ge 7$, $\left|BC\right|\ge \left|B\right|+5$. Moreover, if $〈C〉$ is not isomorphic to the Klein bottle group, i.e. the group with the presentation $〈x,y|xyx=y〉$, then for all subsets $B$ of $G$ with $\left|B\right|\ge 5$, $\left|BC\right|\ge \left|B\right|+5$. The support of an element $\alpha ={\sum }_{x\in G}{\alpha }_{x}x$ in a group algebra $𝔽[G]$ ($𝔽$ is any field), denoted by $\mathrm{\text{supp}}(\alpha )$, is the set $\{x\in G|{\alpha }_x\ne 0\}$. By the latter result, we prove that if $\alpha \beta =0$ for some nonzero $\alpha ,\beta \in 𝔽[G]$ such that $|\mathrm{\text{supp}}(\alpha )|=3$, then $|\mathrm{\text{supp}}(\beta )|\ge 12$. Also, we prove that if $\alpha \beta =1$ for some $\alpha ,\beta \in 𝔽[G]$ such that $|\mathrm{\text{supp}}(\alpha )|=3$, then $|\mathrm{\text{supp}}(\beta )|\ge 10$. These results improve a part of results in Schweitzer [J. Group Theory 16(5) (2013) 667–693] and Dykema et al. [Exp. Math. 24 (2015) 326–338] to arbitrary fields, respectively.

In this work, we computate the trace form ${\mathrm{\text{Tr}}}_{𝕂/\mathbb{Q}}(x^2)$ associated to a cyclic number field $𝕂$ of odd prime degree $p$, where $p$ ramified in $𝕂$ and $x$ belongs to the ring of integers of $𝕂$. Furthermore, we use this trace form to calculate the expression of the center density of algebraic lattices constructed via the Minkowski embedding from some ideals in the ring of integers of $𝕂$.