Journal of Algebra and Its Applications

A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.

Let $R$ be a ring and $(S,\le )$ a strictly ordered monoid. The construction of generalized power series ring $R[[S]]$ generalizes some ring constructions such as polynomial rings, group rings, power series rings and Mal’cev–Neumann construction. In this paper, for a reversible right Noetherian ring $R$ and a m.a.n.u.p. monoid $(S,\le )$, it is shown that (i) $R$ is power-serieswise $S$-McCoy, (ii) $R[[S]]$ have Property (A), (iii) $R$ is right zip if and only if $R[[S]]$ is right zip, (iv) $R$ is strongly AB if and only if $R[[S]]$ is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring $R[[S]]$ and the graph-theoretical properties of its undirected zero divisor graph of $\mathrm{\Gamma }(R[[S]])$. A complete characterization for the possible diameters $\mathrm{\Gamma }(R[[S]])$ is given exclusively in terms of the ideals of $R$. Also, we present some examples to show that the assumption “R is right Noetherian” in our main results is not superfluous.

We prove that when Kontsevich’s deformation quantization is applied on weight homogeneous Poisson structures, the operators in the ∗-product formula are weight homogeneous. In the linear Poisson case for a semisimple Lie algebra $𝔤$ the Poisson manifold $X$ is ${𝔤}^{\ast }$. As an application we provide an isomorphism between the Cattaneo–Felder–Torossian reduction algebra $H^0({𝔪}^{\perp },\chi ,𝔮)$ and the $W$-algebra ${(U(𝔤)/U(𝔤){𝔪}_{\chi })}^{𝔪}$. We also show that in the $W$-algebra setting, ${(S(𝔤)/S(𝔤){𝔪}_{\chi })}^{𝔪}$ is polynomial. Finally, we compute generators of $H^0({𝔪}^{\perp },\chi ,𝔮)$ as a deformation of ${(S(𝔤)/S(𝔤){𝔪}_{\chi })}^{𝔪}$.

A ring $R$ is strongly 2-nil-clean if every element in $R$ is the sum of a tripotent and a nilpotent that commute. We prove that a ring $R$ is strongly 2-nil-clean if and only if $R$ is a strongly feebly clean 2-UU ring if and only if $R$ is an exchange 2-UU ring. Furthermore, we characterize strongly 2-nil-clean ring via involutions. We show that a ring $R$ is strongly 2-nil-clean if and only if every element in $R$ is the sum of an idempotent, an involution and a nilpotent that commute.

The purpose of this paper is to provide useful connections between units and zero divisors, by investigating the structure of a class of rings in which Köthe’s conjecture (i.e. the sum of two nil left ideals is nil) holds. We introduce the concept of unit-IFP for the purpose, in relation with the inserting property of units at zero products. We first study the relation between unit-IFP rings and related ring properties in a kind of matrix rings which has roles in noncommutative ring theory. The Jacobson radical of the polynomial ring over a unit-IFP ring is shown to be nil. We also provide equivalent conditions to the commutativity via the unit-IFP of such matrix rings. We construct examples and counterexamples which are necessary to the naturally raised questions.

In this short note, we generalize a recent result of Zhang [A generalization of Baer’s theorem, Comm. Algebra (2017), doi: 10.1080/00927872.2017.1287275].

In our earlier work on a new functor from $E_6$-Mod to $E_7$-Mod, we found a one-parameter ($c$) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type $E_7$ in $27$ variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector $\overrightarrow{a}\in {ℂ}^{27}\backslash \{\overrightarrow{0}\}$, we prove that the space forms an irreducible $E_7$-module for any $c\in ℂ$ if $\overrightarrow{a}$ is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of $E_6$ over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

Let $R$ be a prime ring with the extended centroid $C$, $L$ a noncommutative Lie ideal of $R$ and $g$ a nonzero $b$-generalized derivation of $R$. For $x,y\in R$, let $[x,y]=xy-yx$. We prove that if $[[\cdots [[g(x^{n_0}),x^{n_1}],x^{n_2}],\dots ],x^{n_k}]=0$ for all $x\in L$, where $n_0,n_1,\dots ,n_k$ are fixed positive integers, then there exists $\lambda \in C$ such that $g(x)=\lambda x$ for all $x\in R$ except when $R\subseteq M_2(F)$, the $2\times 2$ matrix ring over a field $F$. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139–152.]

Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup $\mathrm{\text{Sing}}(X)$ of all non-invertible transformations on a set $X$. The categories involved are characterized as the powerset category $\mathcal{𝒫}(X)$ and the category of partitions $\mathrm{\Pi }(X)$. We describe these categories and show how a permutation on $X$ gives rise to a cross-connection. Further, we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to $\mathrm{\text{Sing}}(X)$. We also describe the right reductive subsemigroups of $\mathrm{\text{Sing}}(X)$ with the category of principal left ideals isomorphic to $\mathcal{𝒫}(X)$. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of $\mathrm{\text{Sing}}(X)$.

The aim of this paper is to show that certain subsets, which are defined by commutativity conditions involving derivations, generalized derivations and epimorphisms, coincide with the center in prime or semiprime rings.

A Lie ideal of a division ring $A$ is an additive subgroup $L$ of $A$ such that the Lie product $[l,a]=la-al$ of any two elements $l\in L,a\in A$ is in $L$ or $[l,a]\in L$. The main concern of this paper is to present some properties of Lie ideals of $A$ which may be interpreted as being dual to known properties of normal subgroups of $A^{\ast }$. In particular, we prove that if $A$ is a finite-dimensional division algebra with center $F$ and $\mathrm{\text{char}}F\ne 2$, then any finitely generated $\mathbb{Z}$-module Lie ideal of $A$ is central. We also show that the additive commutator subgroup $[A,A]$ of $A$ is not a finitely generated $\mathbb{Z}$-module. Some other results about maximal additive subgroups of $A$ and $M_n(A)$ are also presented.

For a *-ring $A$, we associate a simple undirected graph ${\mathrm{\Gamma }}^{*}(A)$ having all nonzero left zero-divisors of $A$ as vertices and, two vertices $x$ and $y$ are adjacent if $x{y}^{*}=0$. In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having ${\mathrm{\Gamma }}^{*}(A)$ a complete graph or a star graph, and sufficient conditions are obtained for ${\mathrm{\Gamma }}^{*}(A)$ to be connected and also for ${\mathrm{\Gamma }}^{*}(A)$ to be disconnected. For a Rickart *-ring $A$, we characterize the girth of $\mathrm{\text{gr}}({\mathrm{\Gamma }}^{*}(A))$ and prove a sort of Beck’s conjecture.

We define a notion of associative representation for algebras. We prove the existence of faithful associative representations for any alternative, Mal’cev, and Poisson algebra, and prove analogs of Ado-Iwasawa theorem for each of these cases. We construct also an explicit associative representation of the Cayley–Dickson algebra in the matrix algebra $M_8(F).$

In this paper, we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules, we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom–Tsfasman weight.

As an extension of the class of Dedekind domains, we have introduced and studied the class of multiplicatively pinched-Dedekind domains (MPD domains) and the class of Globalized multiplicatively pinched-Dedekind domains (GMPD domains) ([T. Dumitrescu and S. U. Rahman, A class of pinched domains, Bull. Math. Soc. Sci. Math. Roumanie 52 (2009) 41–55] and [T. Dumitrescu and S. U. Rahman, A class of pinched domains II, Comm. Algebra 39 (2011) 1394–1403]). The main interest of this paper is to study GMPD domains that have only finitely many overrings.

Let $R$ be a finite commutative ring. We denote by $zp(R)$ the probability that the multiplication of two randomly chosen elements of $R$ is zero. In this paper, we show that either $zp(R)=1$ or $zp(R)\le \frac{3}{4}$ for any ring $R$ and determine all rings $R$ with $zp(R)=\frac{3}{4}$. Also, we obtain the structures of rings $R$ having maximum or minimum value of $zp$ among all rings with identity of the same size. We characterize all rings $R$ with identity such that $zp(R)\ge \frac{3}{8}$. Finally, we compute $zp(R)$ where $R$ is a PIR local ring.

We extend, to rings with zero divisors, the study of perinormal domains initiated by Epstein and Shapiro. We say that a ring $A$ is perinormal if whenever a ring $B$ situated between $A$ and the total quotient ring of $A$ satisfies going down over $A$, it follows that $B$ is a flat $A$-module.

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $𝔞$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $𝔞$, if and only if, ${Ext}_R^i(R/𝔞,M)$ is finitely generated for all $i\le \mathrm{\text{cd}}(𝔞,M)+1$, whenever $dimR/𝔞=1$. In addition, we show that if $M$ is finitely generated and $H_{𝔞}^i(M)$ are weakly Laskerian for all $i\le t-1$, then $H_{𝔞}^i(M)$ are $𝔞$-cofinite for all $i\le t-1$ and for any minimax submodule $K$ of $H_{𝔞}^t(M)$, the $R$-modules ${Hom}_R(R/𝔞,H_{𝔞}^t(M)/K)$ and ${Ext}_R^1(R/𝔞,H_{𝔞}^t(M)/K)$ are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first $Ext$-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all $𝔞$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.

In this paper, vertex representations of the 2-toroidal Lie superalgebras of type $D(m,n)$ are constructed using both bosonic fields and vertex operators based on their loop algebraic presentation.

The group determinant of a finite abelian group of cardinality $n$ is a homogeneous polynomial in $n$ variables of degree $n$. There is no known simple way to compute its coefficients. We derive a recursive formula for them.