Journal of Algebra and Its Applications

We study the Fano scheme of $k$-planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, linear subspaces of sufficiently high dimension must be contained in a coordinate hyperplane. We use our results on these Fano schemes to obtain a lower bound for the product rank of a linear form. This provides a new lower bound for the product ranks of the $6\times 6$ Pfaffian and $4\times 4$ permanent, as well as giving a new proof that the product and tensor ranks of the $3\times 3$ determinant equal five. Based on our results, we formulate several conjectures.

Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra $L$, in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173–199], it was shown that the ideals of $L$ satisfy the distributive law, a property of Prüfer domains and that $L$ is a multiplication ring, a property of Dedekind domains. In this paper, we first show that $L$ satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers $a,b,c$, $gcd(a,b)\cdot lcm(a,b)=a\cdot b$ and $a\cdot gcd(b,c)=gcd(ab,ac)$. We also show that $L$ satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which $L$ satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals.

Let $R$ be an arbitrary ring and $M$ be a nonzero right $R$-module. In this paper, we introduce the set $T_R(M)=\{m\in M|mI=0$, for some nonzero ideal $I$ of $R\}$ of strong torsion elements of $M$ and the properties of this set are investigated. In particular, we are interested when $T_R(M)$ is a submodule of $M$ and when it is a union of prime submodules of $M$.

In this paper, we study (asymptotic) properties of the ∗-distribution of irreducible characters of finite quantum groups. We proceed in two steps, first examining the representation theory to determine irreducible representations and their powers, then we study the ∗-distribution of their trace with respect to the Haar measure. For the Sekine family, we look at the asymptotic distribution (as the dimension of the algebra goes to infinity).

In this paper, we discuss when an order in a Dedekind domain is maximal.

We study the Castelnuovo–Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\ell$ increases by $\lfloor \frac{\ell }{2}\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to

We provide a characterization of the Arf property in both the numerical duplication of a numerical semigroup and in a member of a family of quotients of the Rees algebra studied in [V. Barucci, M. D’Anna and F. Strazzanti, A family of quotients of the Rees algebra, Commun. Algebra 43(1) (2015) 130–142].

Let $R$ be a commutative ring. In this paper, a class of almost projective modules is introduced. An $R$-module $M$ is said to be almost projective if ${\mathrm{\text{Ext}}}_R^1(M,N)=0$ for any $R_{𝔪}$-module $N$, where $𝔪$ is a maximal ideal of $R$. It is shown that an $R$-module $M$ satisfying that $M_{𝔪}$ is free over $R_{𝔪}$ for any maximal ideal $𝔪$ of $R$ is exactly almost projective. As applications, we characterize almost Dedekind domains in terms of almost projectivity and certain divisibility, respectively.

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree $d$. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree $d$.

The paper is focused on questions when some homological and submodule-chain conditions satisfied by a module $M$ are preserved by the group module $MG$. Namely, it is proved for a group $G$ and an $R$-module $M$ that $M{G}_{RG}$ is flat if and only if $M_R$ is flat, and $M{G}_{RG}$ is artinian if and only if $M_R$ is artinian and $G$ is finite, which are two questions raised by Yiqiang Zhou: On Modules Over Group Rings, Noncommutative Rings and Their Applications LENS July 1-4, 2013.

In this paper, we study the notion of $MV$-modules of fractions with respect to a ⋅-prime ideal $P$. Also the relation between the ideals of $MV$-module $M$ and the ideals of $MV$-module of fractions $M_P$ is studied and it is shown that there is a one to one correspondence between $Q$-prime $A$-ideals of $MV$-module $M$ and $Q_P$-prime $A_P^{\prime }$-ideals of $MV$-module of fractions $M_P$, where $A_P^{\prime }$ is the maximal localization of $A$ at a maximal ⋅-ideal $P$ of $A$ and $Q$ is a ⋅-prime ideal of $A$ such that $Q\subseteq P$.

Let ${𝒜}$ and $𝔄$ be two Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible action of $𝔄$ on ${𝒜}$. Let ${𝒜}⋊𝔄$ be a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. We show that (super) amenability of ${𝒜}⋊𝔄$ implies (super) module amenability of ${𝒜}$ and (super) amenability $𝔄$. We investigate biprojectivity and biflatness of ${𝒜}⋊𝔄$ in the some especial cases. We also give some results related to module biprojectivity and module biflatness of ${𝒜}$, when ${𝒜}⋊𝔄$ is biprojective or biflat.

We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present linear time algorithms to compute the Castelnuovo–Mumford regularity and the Krull dimension of binomial edge ideals of block graphs.

We characterize clean elements in unital and general rings by means of outer inverses. Some special cases, such as both clean and unit-regular elements, or strongly clean elements, are discussed. As an application, we also derive new characterizations of strongly regular elements.

We construct ascending chains of ideals in a commutative Noetherian ring $R$ that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal $J$ in $R$, the Ratliff–Rush reduction number $\tilde{r}(J)$ of $J$ is the smallest positive integer $n$ at which the chain $J^2:J\subseteq \cdots \subseteq J^{n+1}:J^n\subseteq \cdots$ becomes stationary. We construct ideals $J$ so that such a chain reaches an arbitrary long sequence of equalities but $\tilde{r}(J)$ is not being reached yet.

In this paper, we introduce the new construction of quandles. For a group $G$ and a subset $A$ of $G$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that if $Q$ is a quandle such that the natural map $Q\to G_Q$ from $Q$ to the enveloping group $G_Q$ of $Q$ is injective, then $Q$ is the $(G,A)$-quandle for an appropriate group $G$ and a subset $A$ of $G$. Also we introduce the free product of quandles and study this construction for $(G,A)$-quandles. In addition, we classify all finite quandles with enveloping group ${\mathbb{Z}}^2$.

Let $H$ be an atomic monoid. For $x\in H$, let $L(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set ${ℒ}(H)=\{L(x)|x\in H\}$. On the other hand, the elasticity of $x$, denoted by $\rho (x)$, is the quotient $supL(x)/infL(x)$ and the elasticity of $H$ is the supremum of the set $\{\rho (x)|x\in H\}$. The system of sets of lengths and the elasticity of $H$ both measure how far $H$ is from being half-factorial, i.e. $|L(x)|=1$ for each $x\in H$.

Let ${𝒞}$ denote the collection comprising all submonoids of finite-rank free commutative monoids, and let ${{𝒞}}_d=\{H\in {𝒞}|\mathrm{\text{rank}}(H)=d\}$. In this paper, we study the system of sets of lengths and the elasticity of monoids in ${𝒞}$. First, we construct for each $d\ge 2$ a monoid in ${{𝒞}}_d$ having extremal system of sets of lengths. It has been proved before that the system of sets of lengths does not characterize (up to isomorphism) monoids in ${{𝒞}}_1$. Here we use our construction to extend this result to ${{𝒞}}_d$ for any $d\ge 2$. On the other hand, it has been recently conjectured that the elasticity of any monoid in ${𝒞}$ is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in ${{𝒞}}_2$ and for any monoid in ${𝒞}$ whose corresponding convex cone is polyhedral.

In this paper, we examine the Lyubeznik tables of two linked ideals $I$ and $J$ of a complete regular local ring $S$ containing a field. More precisely, we prove that the Lyubeznik tables of two evenly linked ideals $I$ and $J$ are the same when $S/I$ and $S/J$ both satisfy one of the following properties: (1) canonically Cohen–Macaulay, (2) generalized Cohen–Macaulay and (3) Buchsbaum.

Furthermore, we give some conditions for equality of Lyubeznik tables of two linked ideals of dimension 2.

Let $R$ be a domain with its field $Q$ of quotients, $M$ an $R$-module and $n$ a fixed non-negative integer. Then $M$ is called $n$-Matlis cotorsion if ${\mathrm{\text{Ext}}}_R^k(Q,M)=0$ for any integer $k>n$. Also $M$ is said to be $n$-Matlis flat if ${\mathrm{\text{Ext}}}_R^1(M,N)=0$ for any $n$-Matlis cotorsion $R$-module $N$. We proved that $({{ℳ}{ℱ}}_n,{{ℳ}{𝒞}}_n)$ is a complete hereditary cotorsion theory, where ${{ℳ}{ℱ}}_n$ (respectively, ${{ℳ}{𝒞}}_n$) denotes the class of all $n$-Matlis flat (respectively, $n$-Matlis cotorsion) $R$-modules. In this paper, it is proved that $R$ is an $n$-Matlis domain if and only if epic images of $(n-1)$-Matlis cotorsion $R$-modules are again $(n-1)$-Matlis cotorsion if and only if $(n-1)$-Matlis flat $R$-modules are of projective dimension $\le 1$ if and only if ${{ℳ}{𝒞}}_0.gl.\mathrm{\text{dim}}(R)\le n$ if and only if ${{ℳ}{𝒞}}_n.gl\mathrm{\text{dim}}(R)=0$.

According to Albu and Iosif, [2, Definition 1.1] a lattice preradical is a subfunctor of the identity functor on the category ${{ℒ}}_{{ℳ}}$ of linear modular lattices, whose objects are the complete modular lattices and whose morphisms are linear morphisms. In this paper, we describe some big lattices which are isomorphic to the big lattice of lattice preradicals and we study the four classical operations that occur in the lattice of preradicals of modules over a ring $R$, namely, the join, the meet, the product and the coproduct. We show that some results about the lattice of module preradicals can be extended to the lattice of lattice preradicals. In particular, we show the existence of the equalizer, the annihilator, the coequalizer and the totalizer for a lattice preradical $\sigma$, as well as some of their properties.