Journal of Algebra and Its Applications

Let $R$ be a commutative Noetherian ring and $C$ a semidualizing $R$-module. We obtain an exact structure $({{\mathscr{H}}}_C,𝜀)$ and prove that the full subcategory ${{\mathscr{H}}}_C\cap {{𝒢}{ℱ}}_C$ of ${{\mathscr{H}}}_C$ is a Frobenius category with ${{ℱ}}_C^{\mathrm{\text{cot}}}$ the subcategory of projective and injective objects, where ${{\mathscr{H}}}_C=\{M|M\cong C\otimes GwithG\text{ a cotorsion module in the Auslander class}\}$ and ${{𝒢}{ℱ}}_C$ (respectively, ${{ℱ}}_C^{\mathrm{\text{cot}}}$) is the subcategory of $C$-Gorenstein flat (respectively, $C$-flat $C$-cotorsion) $R$-modules. Then the stable category $\underline{{{\mathscr{H}}}_C\cap {{𝒢}{ℱ}}_C}$ of ${{\mathscr{H}}}_C\cap {{𝒢}{ℱ}}_C$ and the singularity category ${𝔻}_{\mathrm{\text{sg}}}({{\mathscr{H}}}_C)$ of ${{\mathscr{H}}}_C$ are also considered. As a consequence, we get that there is a Buchweitz’s equivalence $\underline{{{\mathscr{H}}}_C\cap {{𝒢}{ℱ}}_C}\cong {𝔻}_{\mathrm{\text{sg}}}({{\mathscr{H}}}_C)$ if and only if ${{𝒢}{ℱ}}_C$ is a part of some AB-context.

In this paper, we determine the automorphism groups of a class of Frobenius groups, and then solve that under what condition they are REA-groups. As an application, we construct a type of normal edge-transitive Cayley graph.

We study commutative algebras generated by idempotents with particular emphasis on the number of primitive idempotents. Let $R$ be an integral domain with the field of fractions $K$ and let $A$ be an $R$-algebra which is torsion-free as an $R$-module. We show that if $A$ satisfies the three conditions: $A$ is generated by idempotents over $R$; $K{\otimes }_{R}A$ is countably infinite dimensional over $K$; $A$ has $n$ primitive idempotents for a nonnegative integer $n$, then $A$ is uniquely determined up to $R$-algebra isomorphism. We also consider the case where $A$ has countably many primitive idempotents.

In this paper, the concept of almost projective covers is introduced. Using this concept, locally perfect rings are characterized. Also, the concepts of $w$-almost projective modules, $w$-almost projective covers, and $w$-locally perfect rings are introduced. Then $w$-locally perfect rings are characterized in terms of $w$-almost projective modules and $w$-almost projective covers, respectively.

In this paper, we study some algebras $A$ whose unit groups ${𝒰}(A)$ or subnormal subgroups of ${𝒰}(A)$ are (generalized) Engel. For example, we show that any generalized Engel subnormal subgroup of the multiplicative group of division rings with uncountable centers is central. Some of algebraic structures of Engel subnormal subgroups of the unit groups of skew group algebras over locally finite or torsion groups are also investigated.

We show that the image of the representation of the modular group $\mathrm{\text{SL}}(2,\mathbb{Z})$ arising from the representation category $\mathrm{\text{Rep}}(D(G))$ of the Drinfeld double $D(G)$ of a finite abelian group $G$ of exponent $n$ is isomorphic to the special linear group $\mathrm{\text{SL}}(2,\mathbb{Z}/n\mathbb{Z})$, where $\mathbb{Z}/n\mathbb{Z}$ denotes the ring of integers modulo $n$. As a consequence, we establish that the kernel of the representation in question is the principal congruence subgroup of level $n$.

For any group $G$, let $\mathrm{\text{nacent}}(G)$ denote the set of all non-abelian centralizers of $G$. Jafarian Amiri and Rostami, Groups with a few nonabelian centralizers, Publ. Math. Debrecen 87(3–4) (2015) 429–437 put forward the following question: Let $H$ and $G$ be finite simple groups. Is it true that if $|\mathrm{\text{nacent}}(H)|=|\mathrm{\text{nacent}}(G)|$, then $G$ is isomorphic to $H$? In this paper, among other things, we give a negative answer to this question.

In characteristic two, we determine all possible total decompositions of a totally decomposable algebra with orthogonal involution into tensor products of quaternion algebras with involution.

Let $R$ be a commutative Noetherian ring and let $I$ and $J$ be ideals of $R$. The main purpose of this paper is to compare of the finiteness properties of $H_I^i(R)$ and $H_I^i(R\bowtie J)$, where $H_I^i$ is the $i$th local cohomology module functor with respect to $I$ and $R\bowtie J$ is the amalgamated of $R$ along $J$.

Suppose that $R$ is a commutative ring with identity, $I$, $J$ are ideals of $R$, and let $M$ be a finitely generated $R$-module. Let $H_{I,J}^i(-)$ be the $i$th local cohomology functor with respect to $(I,J)$. In this paper, for fixed integers $t$ and $j$, we study the existence of the following isomorphisms of local cohomology modules:

• (i)$H_{I,J}^t(M/xM)\cong H_{I,J}^t(M)\oplus H_{I,J}^{t+1}(M)$;
• (ii)${\mathrm{\text{Ext}}}_R^j(\frac{R}{I},H_{I,J}^t(M/xM))\cong {\mathrm{\text{Ext}}}_R^j(\frac{R}{I},H_{I,J}^t(M))\oplus {\mathrm{\text{Ext}}}_R^j(\frac{R}{I},H_{I,J}^{t+1}(M))$; and,
• (iii)${\mathrm{\text{Tor}}}_j^R(\frac{R}{I},H_{I,J}^t(M/xM))\cong {\mathrm{\text{Tor}}}_j^R(\frac{R}{I},H_{I,J}^t(M))\oplus {\mathrm{\text{Tor}}}_j^R(\frac{R}{I},H_{I,J}^{t+1}(M)),$

for some filter regular element $x$ on $M$. Moreover, we provide some applications of the above isomorphisms.

A right module $M$ is extending if every submodule is essential in a direct summand of $M$. In this note, we obtain a characterization of the right extending generalized triangular matrix rings. This answers a question which was raised in “E. Akalan, G. F. Birkenmeier and A. Tercan, Characterizations of extending modules and ${𝒢}$-extending generalized triangular matrix rings, Comm. Algebra 40 (2012) 1069–1085”.

Let $R$ be a commutative ring with identity and $I^{\ast }(R),$ the set of all nontrivial proper ideals of $R$. The intersection graph of ideals of $R$, denoted by ${ℐ}(R)$, is a simple undirected graph with vertex set as the set $I^{\ast }(R)$, and, for any two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\ne (0)$. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring $R$, ${ℐ}(R)$ is a projective graph if and only if $R\cong R_1\times F_1,$ where $R_1$ is a local principal ideal ring with maximal ideal ${𝔪}_1$ of nilpotency three and $F_1$ is a field. Furthermore, it is shown that for an Artinian ring $R,$ $\overline{\gamma }({ℐ}(R))=2$ if and only if $R\cong R_1\times R_2,$ where each $R_i(1\le i\le 2)$ is a local principal ideal ring with maximal ideal ${𝔪}_i\ne (0)$ such that ${𝔪}_i^2=(0).$

In this paper, we compute the dimension of the Hochschild cohomology groups of any $m$-cluster tilted algebra of type $\widetilde{𝔸}$. Moreover, we give conditions on the bounded quiver of an $m$-cluster tilted algebra $\mathrm{\Lambda }$ of type $\widetilde{𝔸}$ such that the Gerstenhaber algebra ${HH}^{\ast }(\mathrm{\Lambda })$ has nontrivial multiplicative structures. We also show that the derived class of gentle $m$-cluster tilted algebras is not always completely determined by the dimension of the Hochschild cohomology.

Let ${𝒜}$ and $𝔄$ be two $C^{\ast }$-algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible action of $𝔄$ on ${𝒜}$. We define ${𝒜}⋊𝔄$ as a $C^{\ast }$-algebra, where it is a strongly splitting $C^{\ast }$-algebra extension of $𝔄$ by ${𝒜}$. Normal, self-adjoint, unitary, invertible and projection elements of ${𝒜}⋊𝔄$ are characterized; sufficient and necessary conditions for existing unit and bounded approximate identity of ${𝒜}⋊𝔄$ as a Banach algebra and as a $C^{\ast }$-algebra are given. We characterize ∗-automorphisms on ${𝒜}⋊𝔄$ and give some results related to ∗-homomorphisms, ∗-representations and completely bounded maps on this $C^{\ast }$-algebra. Also, we have constructed a new Hilbert $C^{\ast }$-module $X⋊Y$ over ${𝒜}⋊𝔄$, where $X$ is a Hilbert $C^{\ast }$-module over ${𝒜}$ and $Y$ is a Hilbert $C^{\ast }$-module over $𝔄$.

A subgroup $A$ of a group $G$ is called tcc-subgroup in $G$, if there is a subgroup $T$ of $G$ such that $G=AT$ and for any $X\le A$ and for any $Y\le T$, there exists an element $u\in 〈X,Y〉$ such that $X{Y}^u\le G$. The notation $H\le G$ means that $H$ is a subgroup of a group $G$. In this paper, we proved the supersolubility of a group $G=AB$ in the following cases: $A$ and $B$ are supersoluble tcc-subgroups in $G$; all Sylow subgroups of $A$ and of $B$ are tcc-subgroups in $G$; all maximal subgroups of $A$ and of $B$ are tcc-subgroups in $G$. Besides, the supersolubility of a group $G$ is obtained in each of the following cases: all maximal subgroups of every Sylow subgroup of $G$ are tcc-subgroups in $G$; every subgroup of prime order or 4 is a tcc-subgroup in $G$; all 2-maximal subgroups of $G$ are tcc-subgroups in $G$.

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.

It was shown by Galovich that if $R$ is a commutative unique factorization ring (UFR) with identity, then $R$ is a local ring with a nil maximal ideal. In this paper, we generalize Galovich’s results to the non-commutative case.

Let $R$ be a unital simple ring. Under a mild technical restriction on $R$, we characterize bilinear mappings $G:R^2\to R$ satisfying $G(u,u)u^k=u^{k}G(u,u)$, and $G(1,r)=G(r,1)=r$ for all unit $u\in R$ and $r\in R$, where $k\in \mathbb{N}$. As an application we describe bijective linear maps $𝜃:R\to R$ satisfying $𝜃(x{y}^k{x}^{-1}{y}^{-k})=𝜃(x)𝜃{(y)}^k𝜃{(x)}^{-1}𝜃{(y)}^{-k}$ for all invertible $x,y\in R$. Precisely, we will show that $𝜃$ is an isomorphism.

Bounds on the Castelnuovo–Mumford regularity of the associated graded modules of $k$-Buchsbaum modules $M$ are given in terms of $k$ and some other invariants of $M$.

Let $A\subseteq B$ be a commutative ring extension such that $B$ is a trivial extension of $A$ (denoted by $A\propto E$) or an amalgamated duplication of $A$ along some ideal of $A$ (denoted by $A\bowtie I)$. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between $A$ and $B$. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of $B\times C$ by a C-module $E$, where $B$ is an SSP-ring, $C$ a von Neumann regular ring and $E$ a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.

This paper defines the Euler–Poincaré characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [Thanh and Viet, Mixed multiplicities of maximal degrees, J. Korean Math. Soc. 55(3) (2018) 605–622] seem to show a natural and nice picture of mixed multiplicities of maximal degrees.