Journal of Algebra and Its Applications

In this paper, we characterize semirings over which classes of projective, strongly projective, free, and injective semimodules are axiomatizable. Together with injectivity, we consider the concepts of Baer-injectivity and e-injectivity for semimodules over semirings and illustrate possible relationships between axiomatizabilities of the corresponding injective classes of semimodules, as well as characterize semirings over which the class of e-injective semimodules is axiomatizable.

In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials admitting at least one positive real zero in terms of their Newton polytopes and combinatorial structure. Moreover, a class of polynomials attaining their global minimums in the first quadrant are given, which is useful in polynomial optimization.

Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ is complete graph then we describe the generators of the symbolic powers of $I$ and compute the Waldschmidt constant and the resurgence of $I$. Moreover for complete graph we prove that the Castelnuovo–Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if $R$ is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then $R$ is right Noetherian. Also, we show a von Neumann regular (semiregular) ring $R$ with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.

We prove that in any $J$-Noetherian Bezout domain which is not of stable range 1, there exists a nonunit adequate element (element of almost stable range 1).

Let $(K,\nu )$ be a valued field, where $\nu$ is a rank-one discrete valuation, with valuation ring $R$. The goal of this paper is to investigate some basic concepts of Newton polygon techniques of a monic polynomial $f(x)\in R[x]$; namely, theorem of the product, of the polygon, and of the residual polynomial, in such way that improves that given in [D. Cohen, A. Movahhedi and A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) 173–196] and generalizes that given in [J. Guardia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364(1) (2012) 361–416] to any rank-one valued field.

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. Then Rejali and Yousofzadeh introduced the notion of two-sided configuration to study the normal subsets of a group. In [A, Abdollahi, A. Rejali and G. A. Willis, Group properties characterised by configurations, Illinois J. Math. 48(3) (2004) 861–873.], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if $G$ and $H$ have same two-sided configuration sets and $N$ is a normal subgroup of $G$ with polycyclic or FC quotient, then there is a normal subgroup $𝔑$ of $H$ such that $G/N\cong H/𝔑$. Also, we show that if $G$ and $H$ are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic.

Let $G$ be a finite solvable group and let $P\in {\mathrm{\text{Syl}}}_p(G)$ for some prime $p$. Whenever $|G:N_G(P)|$ is odd, Isaacs described a correspondence between irreducible characters of degree not divisible by $p$ of $G$ and $N_G(P)$. This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. In the case where $N_G(P)=P$, G. Navarro showed that every irreducible character $\chi$ of degree not divisible by $p$ has a unique linear constituent ${\chi }^{\ast }$ when restricted to $P$, and that the map $\chi \mapsto {\chi }^{\ast }$ defines a bijection. Navarro’s bijection is obviously natural in the sense described above. We show that these two correspondences are the same under the intersection of the hypotheses.

In this paper, we study 3-Thue–Morse groups, but these are the groups satisfying the semigroup identity $x{y}^{2}xy{x}^{2}y=y{x}^{2}yx{y}^{2}x$. We prove that if $G$ is a 3-Thue–Morse group then $〈x,x^y〉$ is soluble for every $x$ and $y$ in $G$. Furthermore, if $G$ is an Engel group without involution then we show that $G$ is locally nilpotent.

Let $f:{𝕂}^n\to {𝕂}^m$ be a generically finite polynomial map of degree $d$ between affine spaces. In [Z. Jelonek and M. Lasoń, Quantitative properties of the non-properness set of a polynomial map, Manuscripta Math. 156(3–4) (2018) 383–397] we proved that if $𝕂$ is the field of complex or real numbers, then the set $S_f$ of points at which $f$ is not proper is covered by polynomial curves of degree at most $d-1$. In this paper, we generalize this result to positive characteristic. We provide a geometric proof of an upper bound by $d$.

The notion of $f$-ideal was introduced in [G. Q. Abbasi, S. Ahmad, I. Anwar and W. A. Baig, $f$-Ideals of degree 2, Algebra Colloq. 19(1) (2012) 921–926] in 2012 and has been studied in many papers after that. In this paper, we have studied those graphs whose Stanley–Reisner ideals turn out to be $f$-ideals. We give a characterization and construction of these graphs and show that, unlike $f$-graphs, these graphs are always connected. We have also discussed when these graphs are complete bipartite graphs. Moreover, we classify those graphs for which both the facet ideal (the edge ideal) and the Stanley–Resiner ideal, are $f$-ideals.

The aim of the paper is to extend the class of generalized Weyl algebras (GWAs) to a larger class of rings (they are also called GWAs) that are determined by two ring endomorphisms rather than one as in the case of ‘old’ GWAs. A new class of rings, the diskew polynomial rings, is introduced that is closely related to GWAs (they are GWAs under a mild condition). Simplicity criteria are given for GWAs and diskew polynomial rings.

Let ${ℳ}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in ${ℳ}$. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding–Chen rings is given.

We count the number of solutions to a power trace function equal to a constant and use this to find the probability of a successful attack on an authentication code proposed by Ding et al. (2005) [C. Ding, A. Salomaa, P. Solé and X. Tian, Three constructions of authentication/secrecy codes, J. Pure Appl. Algebra 196 (2005) 149–168].

Let $R$ be a semiprime ring, not necessarily with unity, and $a,b\in R$. Let $I(a)$ (respectively, $\mathrm{\text{Ref}}(a)$) denote the set of inner (respectively, reflexive) inverses of $a$ in $R$. It is proved that if $I(a)\cap I(b)\ne \varnothing$, then $I(a)\subseteq I(b)$ if and only if $b=awb=bwa$ for all $w\in I(a)$. As an immediate consequence, if $\varnothing \ne I(a)=I(b)$, then $a=b$ (see Theorem 7 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl. 18(7) (2019) 1950128] for rings with unity). We also give a generalization of Theorem 10 in [A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl. 18(7) (2019) 1950128] by proving that if $\varnothing \ne \mathrm{\text{Ref}}(a)\subseteq \mathrm{\text{Ref}}(b)$ then $a=b$.

Let $G$ be a finite group and let $P_A(G)$ denote the probability that a randomly chosen element from $\mathrm{\text{Aut}}(G)$ fixes a randomly chosen element from $G$. We classify all finite abelian groups $G$ such that $P_A(G)=1/p$ in the cases when $p$ is the smallest prime dividing $|\mathrm{\text{Aut}}(G)|$, and when $p$ is any prime. We also compute $P_A(G)$ for some classes of finite groups. As a consequence of our results, we deduce that if $G$ is a finite $p$-group having a cyclic maximal subgroup, then $\left|G\right|$ divides $|\mathrm{\text{Aut}}(G)|$.

Let $R$ be a commutative ring with $1\ne 0$. Let $n\ge 1$ be a positive integer. A proper ideal $I$ of $R$ is called an n-absorbing ideal (respectively, a strongly n-absorbing ideal) of $R$ as in [D. F. Anderson and A. Badawi, On $n$-absorbing ideals of commutative rings, Comm. Algebra 39 (2011) 1646–1672] if $a_1,a_2,\dots ,a_{n+1}\in R$ and $a_1.a_2\dots a_{n+1}\in I$, then there are $n$ of the $a_i$’s whose product is in $I$ (respectively, if whenever $I_1\dots I_{n+1}\subseteq I$ for ideals $I_1,\dots ,I_{n+1}$ of $R$, then the product of some $n$ of the $I_j^{\prime }$s is contained in $I$). The concept of $n$-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of $R$ is a 1-absorbing ideal of $R$). Let $f:A\to B$ be a ring homomorphism and let $J$ be an ideal of $B.$ This paper investigates the $n$-absorbing and strongly $n$-absorbing ideals in the amalgamation of $A$ with $B$ along $J$ with respect $f$ denoted by $A{\bowtie }^{f}J.$ The obtained results generate new original classes of $n$-absorbing and strongly $n$-absorbing ideals.

This paper seeks an answer to the following question: Let $R$ be a Noetherian ring with $\mathrm{\text{depth}}(R)\le 1$. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain $R$ with $\mathrm{\text{depth}}(R)\le 1$, every ideal is isomorphic to a trace ideal if and only if either $R$ is a DVR or $R$ is one-dimensional divisorial domain, $M$ is a principal ideal of $M^{-1}$ and $M^{-1}$ posses the property that every ideal of $M^{-1}$ is isomorphic to a trace ideal of $M^{-1}$. Next, we globalize our result by showing that a Noetherian domain $R$ with $\mathrm{\text{depth}}(R)\le 1$ has every ideal isomorphic to a trace ideal if and only if either $R$ is a PID or $R$ is one-dimensional divisorial domain, every invertible ideal of $R$ is principal and for every non-invertible maximal ideal $M$ of $R$, $M$ is a principal ideal of $M^{-1}$ and every ideal of $M^{-1}$ is isomorphic to a trace ideal of $M^{-1}$. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

Let $I$ be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume $I$ is contained in the integral closure of some ideal that is generated by two elements from the generating set of $I$. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal $I$. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.