Journal of Algebra and Its Applications

In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective.

In this paper, we established the structure of unitary unit subgroup $V_{\ast }({𝔽}_{2^n}{(QD)}_{16})$ of the group algebra ${𝔽}_{2^n}(Q{D}_{16})$, where $Q{D}_{16}$ is the Quasi-dihedral [D. S. Dummit and R. Foote, Abstract Algebra, 3rd edn. (Wiley, 2004), pp. 71–72] (Semi-Dihedral [B. Huppert, Endliche Gruppen (Springer, 1967), pp. 90–93]) group of order 16 and ${𝔽}_{2^n}$ is any finite field of characteristic 2 with $2^n$ elements.

For a ring $R$ with a derivation $\delta$, we determine the Jacobson radical of the pseudo-differential operator ring $R((x^{-1};\delta ))$ for a large class of rings with a $\delta$-condition. Various types of examples of these rings are provided.

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Let $R,T$ be commutative rings with identity such that $R\subseteq T$. We recall that $R\subseteq T$ is called a $\lambda$-extension of rings if the set of all subrings of $T$ containing $R$ (the “intermediate rings”) is linearly ordered under inclusion. In this paper, a characterization of integrally closed $\lambda$-extension of rings is given. For example, we show that if $R$ is a local ring, then $R\subseteq T$ is an integrally closed $\lambda$-extension of rings if and only if there exists $𝔮\in \mathrm{\text{Spec}}(R)$ such that $T=R_{𝔮},𝔮=T𝔮$ and $R/𝔮$ is a valuation domain. Let $R$ be a subring of $T$ such that $R$ is invariant under action by $G$, where $G$ is a subgroup of the automorphism group of $T$. If $R\subseteq T$ is a $\lambda$-extension of rings, then $R^G\subseteq T^G$ is a $\lambda$-extension of rings under some conditions.

In earlier work, it was established that for any finite field $k$ and any nonempty set $X$, the free associative (nonunitary) $k$-algebra on $X$, denoted by $k{〈X〉}_0$, had infinitely many maximal $T$-spaces, but exactly two maximal $T$-ideals (each of which was shown to be a maximal $T$-space). This raises the interesting question as to whether or not the maximal $T$-spaces can be classified. However, aside from the two maximal $T$-ideals, no examples of maximal $T$-spaces of $k{〈X〉}_0$ have been identified to this point. This paper presents, for each finite field $k$, an infinite set of proper $T$-spaces $W_n(X)$ of $k{〈X〉}_0$, none of which is a $T$-ideal. It is proven that for any distinct integers $r,s\ge 0$, $W_{2^r}(X)+W_{2^s}(X)=k{〈X〉}_0$. Furthermore, it is proven that for the prime field $k={𝔽}_p$, $p$ any prime, $W_1(X)$ is a maximal $T$-space of $k{〈X〉}_0$. We conjecture that for any finite field $k$ of positive characteristic different from 2 and each integer $r\ge 0$, $W_{2^r}(X)$ is a maximal $T$-space of $k{〈X〉}_0$. In characteristic 2, the situation is slightly different and we provide different candidates for maximal $T$-spaces.

Let $G$ be any group and $G$ be a subgroup of $\mathrm{\text{Sym}}(\mathrm{\Omega })$ for some set $\mathrm{\Omega }$. The $2$-closure of $G$ on $\mathrm{\Omega }$, denoted by $G^{(2),\mathrm{\Omega }}$, is by definition,

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, $\nu$-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield${}^{†}$$F$, we pass to the semifield${}^{†}$$F({\lambda }_1,\dots ,{\lambda }_n)$ of fractions of the polynomial semiring${}^{†}$, for which there already exists a well developed theory of kernels, which are normal convex subgroups of $F({\lambda }_1,\dots ,{\lambda }_n)$; the parallel of the zero set now is the $1$-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to $\nu$-kernels (Definition 4.1.4) and $1^{\nu }$-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The $\nu$-kernels corresponding to tropical hypersurfaces are the $1^{\nu }$-sets of what we call “corner internal rational functions,” and we describe $\nu$-kernels corresponding to “usual” tropical geometry as $\nu$-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of $\nu$-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between $1^{\nu }$-sets and a class of $\nu$-kernels of the rational $\nu$-semifield${}^{†}$ called polars, originating from the theory of lattice-ordered groups. When $F$ is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal $\nu$-kernels, intersected with the $\nu$-kernel generated by $F$. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of $\nu$-kernels.

This paper is a sequel to [L. Positselski, Dedualizing complexes and MGM duality, J. Pure Appl. Algebra 220(12) (2016) 3866–3909, arXiv:1503.05523 [math.CT]; Contraadjusted modules, contramodules, and reduced cotorsion modules, preprint (2016), arXiv:1605.03934 [math.CT]]. We extend the classical Harrison–Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules and contramodules over a Matlis domain. This generalizes to the case of any commutative ring $R$ with a fixed multiplicative system $S$ such that the $R$-module $S^{-1}R$ has projective dimension $1$. The latter equivalence connects complexes of $R$-modules with $S$-torsion and $S$-contramodule cohomology modules. It takes a nicer form of an equivalence between the derived categories of abelian categories when $S$ consists of nonzero-divisors or the $S$-torsion in $R$ is bounded.

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen–Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.

We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommuting weak inverses, in semigroups and general rings. In particular, we provide various isomorphisms between the (one-sided, commuting and bicommuting) weak inverses of $ab$ and those of $ba$ (or $ab$ and $cd$ when $ab$ and $cd$ satisfy additional identities).

The non-commuting graph associated to a non-abelian group $G$, ${\mathrm{\Gamma }}_G$, is a graph with vertex set $G\setminus Z(G)$ where distinct non-central elements $x$ and $y$ of $G$ are joined by an edge if and only if $xy\ne yx$. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures:

Conjecture 1. Let $G$ and $H$ be two non-abelian finite groups such that ${\mathrm{\Gamma }}_G\cong {\mathrm{\Gamma }}_H$. Then $\left|G\right|=\left|H\right|$.

Conjecture 2 (AAM’s Conjecture). Let $P$ be a finite non-abelian simple group and $G$ be a group such that ${\mathrm{\Gamma }}_G\cong {\mathrm{\Gamma }}_P$. Then $G\cong P$.

Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824].

In this paper, we will define the same concept for a finite non-commutative Moufang loop $M$ and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

Let $(S,+,\cdot )$ be a semiring generated by one element. Let us denote this element by $w\in S$ and let $g(x)\in x\cdot \mathbb{N}[x]$ be a polynomial. It has been proved that if $g(x)$ contains at least two different monomials, then the elements of the form $g(w)$ may possibly be contained in any countable commutative semigroup. In particular, divisibility of such elements does not imply their torsion. Let, on the other hand, $g(x)$ consist of a single monomial (i.e. $g(x)=k{x}^n$, where $k,n\in \mathbb{N}$). We show that in this case, the divisibility of $g(w)$ by infinitely many primes implies that $g(w)$ generates a group within $(S,+)$. Further, an element $a\in S$ is called an $m$-fraction of an element $z\in S$ if $m\in \mathbb{N}$ and $z=m\cdot a$. We prove that “almost every” $m$-fraction of $w^n$ can be expressed as $f(w)$ for some polynomial $f\in x\cdot \mathbb{N}[x]$ of degree at most $n$.

In this paper, we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide nontrivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

The main scope of this paper is to introduce the valuation semirings in general and discrete valuation semirings in particular. In order to do that, first, we define valuation maps and investigate them. Then we define valuation semirings with the help of valuation maps and prove that a multiplicatively cancellative semiring is a valuation semiring if and only if its ideals are totally ordered by inclusion. We also prove that if the unique maximal ideal of a valuation semiring is subtractive, then it is integrally closed. We end this paper by introducing discrete valuation semirings and show that a semiring is a discrete valuation semiring if and only if it is a multiplicatively cancellative principal ideal semiring possessing a nonzero unique maximal ideal. We also prove that a discrete valuation semiring is a Gaussian semiring if and only if its unique maximal ideal is subtractive.

Suppose the ground field $𝔽$ is an algebraically closed field of characteristic zero. By means of spectral sequences, the computation of the first cohomology group of the model filiform Lie superalgebra $L_{n,m}$ with coefficients in the adjoint module is reduced to the computation of the first cohomology group of an Abel ideal and a one-dimensional subalgebra. Then, by calculating the outer derivations, the algebra structure of the first cohomology group of $L_{n,m}$ is completely characterized.

A ring $R$ is called left comorphic if, for each $a\in R,$ there exists $b\in R$ such that $Ra=l(b)$ and $r(a)=bR.$ Examples include (von Neumann) regular rings, and ${\mathbb{Z}}_{p^n}$ for a prime $p$ and $n\ge 1.$ One motivation for studying these rings is that the comorphic rings (left and right) are just the quasi-morphic rings, where $R$ is left quasi-morphic if, for each $a\in R,$ there exist $b$ and $c$ in $R$ such that $Ra=l(b)$ and $l(a)=Rc.$ If $b=c$ here the ring is called left morphic.

It is shown that $R$ is left comorphic if and only if, for any finitely generated left ideal $L\subseteq R,$ there exists $b\in R$ such that $L=l(b)$ and $r(L)=bR.$ Using this, we characterize when a left comorphic ring has various properties, and show that if $R$ is local with nilpotent radical, then $R$ is left comorphic if and only if it is right comorphic. We also show that a semiprime left comorphic ring $R$ is semisimple if either $R$ is left perfect or $R$ has the ACC on $\{r(x)|x\in R\}.$

After a preliminary study of left comorphic rings with the ACC on $\{l(x)|x\in R\},$ we show that a quasi-Frobenius ring is left comorphic if and only if every right ideal is principal; if and only if every left ideal is a left principal annihilator. We characterize these rings as follows: The following are equivalent for a ring $R:$

• $R$ is quasi-Frobenius and left comorphic.
• $R$ is left comorphic, left perfect and right Kasch.
• $R$ is left comorphic, right Kasch, with the ACC on $\{l(x)|x\in R\}.$
• $R$ is left comorphic, left mininjective, with the ACC on $\{l(x)|x\in R\}.$

Some examples of these rings are given.

For certain metabelian $p$-groups $G=〈x,y〉$ with two generators $x$ and $y$, the annihilator $𝔄⊴\mathbb{Z}[X,Y]$ of the main commutator $[y,x]$ of $G$, as an ideal of bivariate polynomials with integer coefficients, is determined by means of a presentation for $G$. It is proved that together with Schreier’s polynomials $F_x,F_y$, the annihilator $𝔄$ identifies the group $G$ uniquely, and the Furtwängler isomorphism of the additive group underlying the residue class ring $\mathbb{Z}[X,Y]/𝔄$ to the commutator subgroup $G^{\prime }$ of $G$ admits the calculation of the abelian type invariants of $G^{\prime }$. The results are underpinned by class field theoretic realizations of the groups $G$ as Galois groups $\mathrm{\text{Gal}}({\mathrm{\text{F}}}_p^{2}K/K)$ of second Hilbert $p$-class fields ${\mathrm{\text{F}}}_p^{2}K$ over algebraic number fields $K$.

Let $K$ be a field of characteristic zero. For positive integers $n$ and $c$, with $n\ge 2$, let $C_{n,c}$ be a free center-by-metabelian and nilpotent Lie algebra over $K$ of rank $n$ and class $c$, freely generated by a set ${ℒ}$. It is shown that the automorphism group $\mathrm{\text{Aut}}(C_{n,c})$ of $C_{n,c}$ is generated by the general linear group ${\mathrm{\text{GL}}}_n(K)$ and two more IA-automorphisms. Let $K=\mathbb{Q}$ be the field of rational numbers. We give $C_{n,c}$ the structure of a group, say $R_{n,c}$, via the Baker–Campbell–Hausdorff formula. Let $H_{n,c}$ be the subgroup of $R_{n,c}$ generated by ${ℒ}$. We prove that the subgroup of $\mathrm{\text{Aut}}(H_{2,c})$ generated by the tame automorphisms $T_{H_{2,c}}$ and three more IA-automorphisms of $H_{2,c}$ has finite index in $\mathrm{\text{Aut}}(H_{2,c})$. For $n\ge 3$, the subgroup of $\mathrm{\text{Aut}}(H_{n,c})$ generated by the tame automorphisms $T_{H_{n,c}}$ and two more IA-automorphisms of $H_{n,c}$ has finite index in $\mathrm{\text{Aut}}(H_{n,c})$. A similar result is proved for the automorphism group of a free center-by-metabelian and nilpotent group of rank $n$ and class $c$.

Let $R$ be an associative ring. We characterize rings $R$ with simple Lie ring $L\mathrm{\text{Der}}R$ of all Lie derivations, reduced noncommutative Noetherian ring $R$ with the simple Lie ring $\mathrm{\text{Der}}R$ of all derivations and obtain some properties of $2$-torsion-free rings $R$ with the simple Lie ring $J\mathrm{\text{Der}}R$ of all Jordan derivations.