Journal of Algebra and Its Applications

In this paper, we study the zero-divisor graph $\mathrm{\Gamma }(P)$ of a poset $P$ and its line graph $L(\mathrm{\Gamma }(P))$. We characterize all posets whose $\mathrm{\Gamma }(P)$ are star, finite complete bipartite or finite. Also, we prove that the diameter of $L(\mathrm{\Gamma }(P))$ is at most 3 while its girth is either 3, 4 or $\infty$. We also characterize $L(\mathrm{\Gamma }(P))$ in terms of their diameter and girth. Finally, we classify all posets $P$ whose $L(\mathrm{\Gamma }(P))$ are planar.

Let $S$ be a polynomial ring over a field $k$ and $L\subseteq S$ a monomial ideal. We give some inequalities on Stanley regularity of monomial ideals. As consequences, we prove that ${\mathrm{\text{sreg}}}_S(S/L)\le {\mathrm{\text{reg}}}_S(S/L)$ and ${\mathrm{\text{sreg}}}_S(L)\le {\mathrm{\text{reg}}}_S(L)$ hold in the following cases: (1) $L$ is a complete intersection; (2) $L$ is an ideal of mixed products.

This paper classifies irreducible, integrable highest weight modules for “current Kac–Moody Algebras” with finite-dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac–Moody Lie algebras.

Let $\pi :G\to {{𝒢}}_X$ be an isometric representation of a group $G$ in a Banach space $X$ over a normalizing non-discrete absolute valued division ring $K$. If $\pi$ and ${\pi }^{\ast }$ are supportive and $X$ verifies the separation property, then $X^G$ is 1-complemented in $X$ along $X_G$. As an immediate consequence, in an isometric representation of a group in a smooth Banach space whose dual is also smooth, the subspace of $G$-invariant vectors is $1$-complemented.

In this paper, we study symmetric pairs and pseudosymmetries in the Hom-Yetter–Drinfeld category ${}_H^H𝕐𝔻$ over a Hom-Hopf algebra $(H,S,\beta )$. We first show that the category ${}_H^H𝕐𝔻$ over a (co)triangular Hom-Hopf algebra $H$ contains a rich symmetric subcategory. Also we prove that the (co)commutativity and trivial property of $H$ are determined by some symmetric pairs of objects in ${}_H^H𝕐𝔻$. Moreover, we find a sufficient and necessary condition for ${}_H^H𝕐𝔻$ to be pseudosymmetric.

In this paper, we study the atomic structure of the family of Puiseux monoids, i.e. the additive submonoids of ${\mathbb{Q}}_{\ge 0}$. Puiseux monoids are a natural generalization of numerical semigroups, which have been actively studied since mid-19th century. Unlike numerical semigroups, the family of Puiseux monoids contains non-finitely generated representatives. Even more interesting is that there are many Puiseux monoids which are not even atomic. We delve into these situations, describing, in particular, a vast collection of commutative cancellative monoids containing no atoms. On the other hand, we find several characterization criteria which force Puiseux monoids to be atomic. Finally, we classify the atomic subfamily of strongly bounded Puiseux monoids over a finite set of primes.

We prove that all hypergroups of order four are commutative and that there exists a non-comutative hypergroup of order five. These facts imply that the minimum order of non-commutative hypergroups is five, even though the minimum order of non-commutative groups is six.

A ring $R$ is defined to be feebly clean, if every element $x$ can be written as $x=u+e_1-e_2$, where $u$ is a unit and $e_1$, $e_2$ are orthogonal idempotents. Feebly clean rings generalize clean rings and are also a proper generalization of weakly clean rings. The family of all semiclean rings properly contains the family of all feebly clean rings. Further properties of feebly clean rings are studied, some of them analogous to those for clean rings. The feebly clean property is investigated for some rings of complex-valued continuous functions. Throughout, all rings are commutative with identity.

The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements whose squares lie in the square of the underlying field.

For a semimodule over any semiring, the maximal and prime $k$-subsemimodules are characterized with the help of the quotient structure. The main results are as follows. A semimodule is $k$-congruence-simple iff it is $k$-subsemimodule-simple. A nonzero semimodule is $k$-simple iff it satisfies condition $(C)$. A proper $k$-subsemimodule of a semimodule is a maximal $k$-subsemimodule iff the quotient semimodule is $k$-simple. A proper $k$-subsemimodule is prime in a semimodule iff the quotient semimodule is prime. A nonzero $k$-simple semimodule over a commutative semiring is prime. Lemma 6.2 in this paper corrects the statement and proof of Lemma 3.16 in [Yeşilot Hacet. J. Math. Stat. 39 (2010) 305–312].

In this paper, we survey the main known constructions of Ferrers diagram rank-metric codes, and establish new results on a related conjecture by Etzion and Silberstein. We also give a sharp lower bound on the dimension of linear rank-metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters. We also apply results from algebraic geometry to the study of the analogous problem over an algebraically closed field, proving that the bound by Etzion and Silberstein can be improved in this case, and providing a sharp bound for full-rank matrices.

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The essential graph of $R$ is defined as the graph $EG(R)$ with the vertex set $Z{(R)}^{*}=Z(R)\setminus \{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if ann${}_R(xy)$ is an essential ideal. It is proved that $EG(R)$ is connected with diameter at most three and with girth at most four, if $EG(R)$ contains a cycle. Furthermore, rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a ring. Finally, we show that the essential graph associated with an Artinian ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring $R$ and a monoid $M$, we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring $R[M]$. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.

Let $G$ be a finite group. If $K$ is a subgroup of $G$ and $H$ a subgroup of $K$, we say that $H$ is strongly closed in $K$ with respect to $G$ if $K\cap H^g\le H$ for any $g\in G$. We say that a subgroup $H$ of $G$ is strongly closed in $G$ if $H$ is strongly closed in $N_G(H)$ with respect to $G$. A subgroup $H$ of a group $G$ is said to be weakly $c$-supplemented in $G$ if $G$ has a subgroup $K$ such that $G=HK$ and $H\cap K$ is strongly closed in $G$. In this paper, we study the structure of a group $G$ under the assumption, that some subgroups of prime power order are weakly $c$-supplemented in $G$. Our results extend and generalize several recent results in the literature.

An element $a$ of a ring $R$ is nil-clean, if $a=e+b$, where $e^2=e\in R$ and $b$ is a nilpotent element, and the ring $R$ is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring $R$ and an abelian group $G$, the group ring $RG$ is nil-clean, iff $R$ is nil-clean and $G$ is a $2$-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite $2$-group over a nil-clean ring is nil-clean, and that the hypercenter of the group $G$ must be a $2$-group if a group ring of $G$ is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a $2$-group.

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $char(F)=p$ is a finite integer $d$ greater than 1 then the symbol length of $p$-algebras of exponent $p$ over $F$ is bounded from above by $\lceil \frac{d-1}{p}\rceil -1$, and show that every two tensor products of symbol algebras of lengths $k$ and $\ell$ with $(k+\ell )p\ge d-1$ can be modified so that they share a common slot. For $p=2$, we obtain an upper bound of $\frac{u(F)}{2}-1$ for the symbol length, which is sharp when $I_q^{3}F=0$.

In this paper, in looking for the weak Hopf algebraic counterpart of pseudosymmetric braidings, we introduce the concept of a pseudotriangular weak Hopf algebra, which is a quasitriangular weak Hopf algebra satisfying an extra condition. Then, we investigate the question, when a quasitriangular weak Hopf algebra is pseudotriangular. As an application, we study a special class of pseudotriangular weak Hopf algebras, under the name almost-triangular weak Hopf algebras and list some nontrivial examples. Finally, in order to construct more examples of pseudotriangular weak Hopf algebras, we show that the pseudosymmetry of the Yetter–Drinfeld category ${}_H^H{𝒴}{𝒟}$ is determined by the commutativity and cocommutativity of $H$, where $H$ is a weak Hopf algebra with a bijective antipode.

Let ${{𝒫}}_n$ and ${{𝒮}{𝒫}}_n$ be the partial transformation and the strictly partial transformation semigroups on the finite set $X_n=\{1,2,\dots ,n\}$. It is well known that the ranks of the semigroups ${𝒫}(n,r)=\{\alpha \in {{𝒫}}_n:|\mathrm{im}(\alpha )|\le r\}$ and ${𝒮}{𝒫}(n,r)=\{\alpha \in {{𝒮}{𝒫}}_n:|\mathrm{im}(\alpha )|\le r\}$ are $S(n+1,r+1)$, for $2\le r\le n-1$, and $(r+1)S(n,r+1)$, for $2\le r\le n-2$, respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup ${𝒫}(n,r)$ has the same value as the rank. Idempotent can be seen as a special case (with $p=q=1$) of $(p,q)$-potent. In this paper, we determine the $(p,q)$-potent ranks, defined as the smallest number of $(p,q)$-potents generating set, of the semigroups ${𝒫}(n,r)$, for $2\le r\le n-1$, and ${𝒮}{𝒫}(n,r)$, for $2\le r\le n-2$.

In this paper, we continue our research on Koszul rings, started in [P. Jara, J. López-Peña and D. Ştefan, Koszul pairs and applications, to appear in J. Noncommut. Geom., http://arxiv.org/pdf/1011.4243.pdf]. In Theorem 1.9, we prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of ${\mathrm{\text{Tor}}}_n^A(R,R)$. As an application of these characterizations, we investigate the Koszulity of the incidence rings for finite graded posets, see Theorems 2.8 and 2.9. Based on these results, we describe an algorithm to produce new classes of Koszul posets (i.e. graded posets, whose incidence rings are Koszul). Specific examples of Koszul posets are included.

The rings in the title were called UN rings by Călugăreanu in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182]. He gave two examples of simple UN rings: matrix rings over a skew field and a ring, which is the filtered union of such rings. We give new examples of simple UN rings as endomorphism rings of ‘vector space like’ modules and determine the structure of UN rings, which satisfy a polynomial identity or have Krull dimension. We also answer some questions in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182] about Morita equivalence of UN rings and show that this question is related to Köthe’s conjecture. Finally a complete characterization is given of modules over a Dedekind domain (in particular Abelian groups) and modules of finite length with a UN endomorphism ring.