Journal of Algebra and Its Applications

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map $F$ on an affine space of dimension $n$ over a field of characteristic $0$, we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of $F$ as well as a formula for $F^{-1}$ in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

Let $p$ be an odd prime, and let $G$ be a nilpotent group, whose commutator subgroup is finite abelian satisfying $G^{\prime }={\mathrm{\text{Syl}}}_p(G)$ and ${\gamma }_3(G)\subseteq {(G^{\prime })}^p$. In this contribution, an upper bound is given on the derived length of the group of units of the group algebra of $G$ over a field of characteristic $p$. Furthermore, we show that this bound is achieved, whenever $G^{\prime }$ is cyclic.

Let $R$ be a commutative ring and $M$ an $R$-module. Let $Z(M)=\{r\in R|rm=0\mathrm{\text{ for some nonzero }}m\in M\}$ and $T(M)=\{m\in M|rm=0\mathrm{\text{ for some nonzero }}r\in R\}$. $M$ satisfies Property $A$ (respectively, Property $T$) if for each finitely generated ideal $I\subseteq Z(M)$ (respectively, finitely generated submodule $N\subseteq T(M)$) ann${}_M(I)\ne 0$ (respectively, ann${}_R(N)\ne 0$). The ring $R$ satisfies Property $A$ if ${}_{R}R$ does. We study rings and modules satisfying Property $A$ or Property $T$. A number of examples are given, many using the method of idealization.

In the past two decades there has been great attention to Lie (super)algebras, which are extensions of affine Kac–Moody Lie (super)algebras, in certain typical or axiomatic approaches. These Lie (super)algebras have been mostly studied under variations of the name “extended affine Lie (super)algebras”. We show that certain classes of Malcev (super)algebras also can be put in this framework. This in particular allows us to provide new examples of Malcev (super)algebras which extend the known Kac–Moody Malcev (super)algebras.

Let $I$ be the edge monomial ideal of a graph $G$, whose vertex set is $V(G)=\{v_1,\dots ,v_n\}$. $G$ is implosive if the symbolic Rees algebra $R_S(I)$ of $I$ has a minimal system of generators $x^{w_1}{t}^{b_1},\dots ,x^{w_s}{t}^{b_s}$ where $x^{w_1},\dots ,x^{w_s}$ are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under clique-sums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.

In this paper, we introduce and study (weak) pure-injective Gorenstein projective modules. Let $R$ be an Artin algebra. We prove that the category of weak pure-injective Gorenstein projective left $R$-modules coincides with the intersection of the category of pure-injective left $R$-modules and that of Gorenstein projective left $R$-modules. Then, we get an equivalent characterization of virtually Gorenstein algebras (being CM-finite). Furthermore, we prove that the category of weak pure-injective Gorenstein projective left $R$-modules is enveloping in the category of left $R$-modules; and if $R$ is virtually Gorenstein, then it is precovering in the category of pure-injective left $R$-modules.

Let $R$ be an integral domain, $K=qf(R)$ and $F(R)$ the set of fractional ideals of $R$. Let $\mathrm{\text{GV}}(R)=\{I|I$ a finitely generated ideal with $I^{-1}=R\}$. For a torsion-free $R$-module $M$, define $M_w=\{x\in M\underset{R}{\otimes }K|Jx\subseteq M$ for some $J\in \mathrm{\text{GV}}(R)\}$. Call $M$ a $w$-module if $M=M_w$. On $F(R)$, the function $I\to I_w$ is a star-operation of finite character. An integral ideal $P$ maximal with respect to being a proper $w$-ideal is a prime ideal called a maximal $w$-ideal. A torsion-free $R$-module $M$ is called $w$-flat, if $M_P\cong M\underset{R}{\otimes }{R}_P$ is a flat $R_P$-module for each $P\in w\text{-}max(R)$, the set of maximal $w$-ideals of $R$. $R$ is called a Prüfer $v$-multiplication domain (P$v$MD), if $R_M$ is a valuation ring for each $M\in w\text{-}max(R)$. We characterize $w$-flat modules in a manner similar to the characterization of flat modules, study them when they are rings $T$ with $R\subseteq T\subseteq K$ and characterize P$v$MDs using them and compare our work with similar work in the literature.

In this work, we provide a suite of protocols for group key management based on general semigroup actions. Construction of the key is made in a distributed and collaborative way. Examples are provided that may in some cases enhance the security level and communication overheads of previous existing protocols. Security against passive attacks is considered and depends on the hardness of the semigroup action problem in any particular scenario.

We analyze the structure of the virtual (orbifold) $K$-theory ring of the complex orbifold $\mathbb{P}(1,n)$ and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin–Graham [D. Edidin and W. Graham, Nonabelian localization in equivariant $K$-theory and Riemann–Roch for quotients, Adv. Math. 198(2) (2005) 547–582]. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual $K$-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual $K$-theory ring of $\mathbb{P}(1,n)$ to the ordinary $K$-theory ring of a crepant resolution of the cotangent bundle of $\mathbb{P}(1,n)$ which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of $\mathbb{P}(1,n)$ which is isomorphic to the ordinary K-theory ring of the resolution. This generalizes the results of Edidin–Jarvis–Kimura [D. Edidin, T. J. Jarvis and T. Kimura, Chern classes and compatible power operation in inertial $K$-theory, Ann. K-Theory (2016)], who proved the latter for $n=2,3$.

A magma is just a set $A$ with a single binary operation

We investigate the Ihara zeta functions of finite Schreier graphs ${\mathrm{\Gamma }}_n$ of the Basilica group. We show that ${\mathrm{\Gamma }}_{1+n}$ is two sheeted unramified normal covering of ${\mathrm{\Gamma }}_n$, for all $n\ge 1$ with Galois group $\frac{\mathbb{Z}}{2\mathbb{Z}}.$ In fact, for any $n>1,r\ge 1$, the graph ${\mathrm{\Gamma }}_{n+r}$ is $2^n$ sheeted unramified, non-normal covering of ${\mathrm{\Gamma }}_r$. In order to do this, we give the definition of the $generalized$ $replacement$ $product$ of Schreier graphs of Basilica groups. We also show the corresponding results in zig-zag product of Schreier graphs ${\mathrm{\Gamma }}_n$ with a $4$-cycle.

An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring ${𝔽}_q{C}_{p^k}$ is ∗-clean, where ${𝔽}_q$ is a finite field and $C_{p^k}$ is a cyclic group of an odd prime power order $p^k$. They also provided a necessary condition and a few sufficient conditions for ${𝔽}_q{C}_n$ to be ∗-clean, where $C_n$ is a cyclic group of order $n$. In this paper, we extend the above result of Huang, Li and Yuan from ${𝔽}_q{C}_{p^k}$ to $𝔽G$ and provide a characterization of ∗-clean group rings $𝔽G$, where $G$ is a finite abelian group and $𝔽$ is a field with characteristic not dividing the exponent of $G$.

We determine up to isomorphism finite nonabelian $2$-groups all of whose minimal nonabelian subgroups, except one, are isomorphic to $D_8$ (Theorem 1) and also, we determine finite nonabelian $2$-groups all of whose minimal nonabelian subgroups, except one, are isomorphic to $Q_8$ (Theorem 2).

A Hom-structure on a Lie algebra $(𝔤,[,])$ is a linear map $\sigma :𝔤\to 𝔤$ satisfying the Hom–Jacobi identity: $[\sigma (x),[y,z]]+[\sigma (y),[z,x]]+[\sigma (z),[x,y]]=0$ for all $x,y,z\in 𝔤$. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. In this paper, using a classification theorem due to Mathieu, we determine explicitly all the Hom-structures on the simple graded Lie algebras of finite growth. As a direct consequence, all the Hom-structures on any simple graded Lie algebras of finite growth constitute a Jordan algebra in the usual way.

In this paper, we address the Hadamard product of linear varieties not necessarily in general position. In ${\mathbb{P}}^2$, we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets $X$ and $X^{\prime }$ of collinear points, we get conditions for $X\star X^{\prime }$ to be either a collinear finite set of points or a grid of $\left|X\right|\left|X^{\prime }\right|$ points. In ${\mathbb{P}}^3,$ under suitable conditions (which we prove to be generic), we show that $X\star X^{\prime }$ consists of $\left|X\right|\left|X^{\prime }\right|$ points on the two different rulings of a nondegenerate quadric and we compute its Hilbert function in the case $\left|X\right|=\left|X^{\prime }\right|.$

In this note, we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (respectively, right) ZAFS property is shown to pass also to classical left (respectively, right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.

In this note, we study the cohomology of the nonclassical restricted Lie algebras $W(n)$, $n\in {\mathbb{N}}^{+}$ over a field $𝔽$ of characteristic $p>0$, which are by definition the Lie algebras of derivations on the truncated polynomial algebras $𝔽[x_1,\dots ,x_n]/(x_1^p,\dots ,x_n^p)$ (called the Jacobson–Witt algebras), and simple unless $p=2$ and $n=1$. We show a vanishing theorem for the cohomology of $W(n)$ with coefficients in the trivial module $𝔽$, which says that $H^q(W(n))=0$ for all $q=1,\dots ,min\{n,p\}-1$. Moreover, we compute the cohomology of $W(n)$ with coefficients in its defining truncated polynomial algebra under a certain assumption on the characteristic of the ground field.

Gagola and Lewis proved that a finite group $G$ is nilpotent if and only if $\chi {(1)}^2$ divides $|G:$ $\mathrm{\text{Ker}}$ $\chi |$ for all irreducible characters $\chi$ of $G$. In this paper, we prove that a finite soluble group $G$ is nilpotent if and only if $\chi {(1)}^2$ divides $|G:$ $\mathrm{\text{Ker}}$ $\chi |$ for all irreducible monomial characters $\chi$ of $G$.

A $p$-group $G$ is called a $C{C}_t^s$-group if $|G^{\prime }/(G^{\prime }\cap N)|$ $\le p^s$ or $|N/(N\cap Z(G))|\le p^t$ for every normal subgroup $N$ in $G$. In this paper, we first investigate the properties of $C{C}_t^s$-groups, in particular, we give the upper bounds of the exponent of $G^{\prime }$ with $G^{\prime }\le Z(G)$ and the order of $G^{\prime }Z(G)/Z(G)$ for a $C{C}_t^s$-group $G$. Then, we try to describe the structure of $C{C}_1^1$-groups, and a necessary condition for a $p$-group to be a $C{C}_1^1$-group is given. We also give some elementary properties of $p$-groups with very small derived subgroups by using the properties of capable $p$-groups, which maybe could apply some other research of $p$-groups.

Let $H$ be a subgroup of a finite group $G$. We say that $H$ is a $c^{\#}$-normal subgroup of $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is a $\mathrm{\text{CAP}}$-subgroup of $G$. In the present paper, we use $c^{\#}$-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be $p$-supersolvable, $p$-nilpotent and solvable. Our results extend and improve some recent ones.