Journal of Algebra and Its Applications

A family of quotients of a Rees algebra associated to a ring with respect to a fixed ideal was recently introduced by Barucci, D’Anna and Strazzanti. In this paper, we will classify rings of this family that satisfy certain Prüfer-like properties and, as a particular case, we will extend results obtained for amalgamated duplications and Nagata idealizations.

A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967–980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities.

It is well known that a domain $R$ is a Prüfer domain if and only if for $R$-modules, relative divisibility implies purity. In this paper, we extend this result to Prüfer $v$-multiplication domains (P$v$MDs). To do this, we introduce the concept of $w$-pure exact sequences over commutative rings with zero divisors, and we prove that a domain $R$ is a P$v$MD if and only if for $R$-modules, relative divisibility implies $w$-purity. Also, we compare $w$-purity with purity and relative divisibility by giving examples.

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov — M. Kerr and the author — together with D. Hernández Serrano — can also be deduced from this general expression.

Let $I$ and $J$ be two ideals in a commutative Noetherian ring $S$. We say that $J$ is a superficial ideal for $I$ if the following conditions are satisfied:

• ⁽ⁱ⁾$G(J)\subseteq G(I)$, where $G(L)$ denotes a minimal set of generators of an ideal $L$.
• ⁽ⁱⁱ⁾$(I^{k+1}{:}_{S}J)=I^k$ for all positive integers $k$.

In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

As an extension of ${𝔸𝔾}_z(L)$ (the annihilation graph of the commutator poset (lattice) $L$ with respect to an element $z\in L$), we discuss when ${𝔸𝔾}_I(L)$ (the annihilation graph of the commutator poset (lattice) $L$ with respect to an ideal $I\subseteq L$) is a complete bipartite ($r$-partite) graph together with some of its other graph-theoretic properties. We investigate the interplay between some (order-) lattice-theoretic properties of $L$ and graph-theoretic properties of its associated graph ${𝔸𝔾}_I(L)$. We provide some examples to show that some conditions are not superfluous assumptions. We prove and show by a counterexample that the class of lower sets of a commutator poset $L$ is properly contained in the class of $m$-ideals of $L$ (i.e. multiplicatively absorptive ideals (sets) of $L$ that are defined by commutator operation).

Let $A$ be a finite-dimensional hereditary algebra over a finite field, and let $m$ be a fixed integer such that $m=0$ or $m>2$. In the present paper, we first define an algebra $L_m(A)$ associated to $A$, called the $m$-periodic lattice algebra of $A$, and then prove that it is isomorphic to Bridgeland’s Hall algebra ${𝒟}{{\mathscr{H}}}_m(A)$ of $m$-cyclic complexes over projective $A$-modules. Moreover, we show that there is an embedding of the Heisenberg double Hall algebra of $A$ into ${𝒟}{{\mathscr{H}}}_m(A)$.

We give a uniform General Neron Desingularization (GND) for one-dimensional local rings with respect to morphisms which coincide modulo a high power of the maximal ideal. The result has interesting applications in the case of Cohen–Macaulay rings.

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring $R$. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field $k$ into the general setting of algebras over an arbitrary ring $R$. For this sake, we introduce and study the notion of a fibered AF-ring over a ring $R$. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings $(R,A,B)$ consisting of two $R$-algebras $A$ and $B$ such that $A{\otimes }_{R}B\ne \{0\}$, we introduce the inherent notion to $(R,A,B)$ of a $B$-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product $A{\otimes }_{R}B$. As an application, we provide a formula for the Krull dimension of $A{\otimes }_{R}B$ when either $R$ or $A$ is zero-dimensional as well as for the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a fibered AF-ring over the ring of integers $\mathbb{Z}$ with nonzero characteristic and $B$ is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a Boolean ring. Actually, we prove that if $A$ and $B$ are rings such that $A{\otimes }_{\mathbb{Z}}B$ is not trivial and $A$ is a Boolean ring, then dim$(A{\otimes }_{\mathbb{Z}}B)=\text{dim}(\frac{B}{2B})$.

Let $G$ be a finite non-abelian group and $Z(G)$ be its center. For a fixed nonidentity element $g$ of $G$, the $g$-noncommuting graph of $G$, denoted by ${\mathrm{\Delta }}_G^g$, is a simple undirected graph in which its vertices are $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent if $[x,y]\ne g$ and $g^{-1}$. In this paper, we discuss about connectivity of ${\mathrm{\Delta }}_G^g$ and determine all finite non-abelian groups such that their $g$-noncommuting graphs are 1-planar, toroidal or projective.

In this work, impecunious modules are introduced as modules whose injectivity domains are contained in the class of all pure-split modules. This notion gives a generalization of both poor modules and pure-injectively poor modules. Properties involving impecunious modules as well as examples that show the relations between impecunious modules, poor modules and pure-injectively poor modules are given. Rings over which every module is impecunious are right pure-semisimple. A commutative ring over which there is a projective semisimple impecunious module is proved to be semisimple artinian. Moreover, the characterization of impecunious abelian groups is given. It states that an abelian group $M$ is impecunious if and only if for every prime integer $p$, $M$ has a direct summand isomorphic to ${\mathbb{Z}}_{p^n}$ for some positive integer $n$. Consequently, an example of an impecunious abelian group which is neither poor nor pure-injectively poor is given so that the generalization defined is proper.

In this paper, a graded ring is a ring which is the direct sum of a family of its additive subgroups indexed by a nonempty set under the assumption that the product of homogeneous elements is again homogeneous. We study graded special radicals and special radicals of graded rings, but which contain the corresponding Jacobson radicals. There are two versions of this graded radical, which we name the graded over-Jacobson and the large graded over-Jacobson radical. We establish several characterizations of the graded over-Jacobson radical of a graded ring and also prove that the largest homogeneous ideal contained in the corresponding classical radical of a graded ring coincides with the large graded over-Jacobson radical of that ring.

A ring $R$ is defined to be 2-nil-good if every element in $R$ is the sum of two units and a nilpotent. Fundamental properties of such rings are obtained. We prove that every strongly $\pi$-regular ring is 2-nil-good if and only if the identity is the sum of two units. One of the main result of this paper is that every square matrix ring over J-fine rings is 2-nil-good. We establish 2-nil-good property for Morita contexts. This implies, in particular, that every matrix ring over 2-nil-good rings is 2-nil-good. Furthermore, we prove that the ring ${𝕃𝕋𝔻𝔽𝕄}_{\mathbb{N}}(R)$ of all lower triangular diagonal-finite matrices over a 2-good ring $R$ is 2-nil-good.

We generalize Ye’s Theorem which states that the group ring ${\mathbb{Z}}_{(p)}[C_3]$ is a semi-clean ring [Y. Ye, Semiclean rings, Comm. Algebra 31 (2003) 5609–5625]. The proof provided here is more efficient; it is less algorithmic but has the feature that the following statement is evident: for distinct primes $p,q$, the group ring ${\mathbb{Z}}_{(p)}{C}_q$ is feebly clean if and only if the order of $p$ modulo $q$ is at least $\frac{q-1}{2}$.

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, $\mathbb{Z}[x]/〈f〉$ (here $f$ is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of them in the multivariate case too. Based on multivariate ideal lattices, we construct hash functions using Gröbner basis techniques. We define a worst case problem, shortest substitution problem with respect to an ideal in $\mathbb{Z}[x_1,\dots ,x_n]$, and use its computational hardness to establish the collision resistance of the hash functions.

The modified quantum algebra $\dot{U}$ associated to a quantum algebra $U$ was introduced by Lusztig. $\dot{U}$ has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of $\dot{U}$ for type $A_2$.

Two isometry groups of combinatorial codes are described: the group of isometries, that is, the group of Hamming isometries from a code to itself and the group of monomial isometries, which is the group of those isometries of a code to itself that extend to monomial maps. Unlike the case of classical linear codes, where these groups are the same, it is shown that for combinatorial codes the groups can be arbitrarily different. In particular, there exist codes with the richest possible group of isometries and the trivial group of monomial isometries. In the paper, the two groups are characterized and codes with predefined isometry groups are constructed.

We provide two sufficient conditions to guarantee that the round functions of a translation-based cipher generate a primitive group. Furthermore, under the same hypotheses, and assuming that a round of the cipher is strongly proper and consists of $m$-bit S-Boxes, with $m=3,4$ or $5$, we prove that such a group is the alternating group. As an immediate consequence, we deduce that the round functions of some lightweight translation-based ciphers, such as the PRESENT cipher, generate the alternating group.

Let $H$ be a subset of a commutative graded ring $R$. The Cayley graph $\mathrm{\text{Cay}}(R,H)$ is a graph whose vertex set is $R$ and two vertices $a$ and $b$ are adjacent if and only if $a-b\in H$. The Cayley sum graph ${\mathrm{\text{Cay}}}^{+}(R,H)$ is a graph whose vertex set is $R$ and two vertices $a$ and $b$ are adjacent if and only if $a+b\in H$. Let $S$ be the set of homogeneous elements and $\mathrm{\text{Z}}(R)$ be the set of zero-divisors of $R$. In this paper, we study ${\mathrm{\text{Cay}}}^{+}(R,\mathrm{\text{Z}}(R))$ (total graph) and $\mathrm{\text{Cay}}(R,S)$. In particular, if $R$ is an Artinian graded ring, we show that $\mathrm{\text{Cay}}(R,S)$ is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of $\mathrm{\text{Cay}}(R,S)$ for some finite graded ring $R$.

We graph-theoretically characterize the class of graphs $G$ such that $I{(G)}^2$ are Buchsbaum.