Journal of Algebra and Its Applications

We investigate the almost Cohen–Macaulay property and the Serre-type condition $(C_n),n\in \mathbb{N},$ for noetherian algebras and modules. More precisely, we find permanence properties of these conditions with respect to tensor products and direct limits.

In this paper, we prove some general results on Leibniz 2-cocyles for simple Leibniz algebras. Applying these results, we prove the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself, whose associated Lie algebra is isomorphic to ${\mathrm{\text{sl}}}_2$.

It is well known in [Absolutely pure modules, Proc. Amer. Math. Soc. 26 (1970) 561–566, Theorem 6] that a domain $R$ is a Prüfer domain if and only if every divisible $R$-module is absolutely pure, where an $R$-module $A$ is called absolutely pure if ${\mathrm{\text{Ext}}}_R^1(N,A)=0$ for every finitely presented $R$-module $N$. In this paper, we extend this result to Prüfer $v$-multiplication domains (P$v$MDs). To do this, comparing with [An Introduction to Homological Algebra, 2nd edn. (Springer, Science+Business Media, LLC, New York, 2009), Theorem 3.69], we firstly give homological characterizations of $w$-purity, and we introduce the concept of absolutely $w$-pure modules over commutative rings with zero divisors. Finally, we prove that a domain $R$ is a P$v$MD if and only if every divisible $R$-module is absolutely $w$-pure, and compare absolutely $w$-purity with absolutely purity by giving an example.

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson’s amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra ${{𝒯}}_2=\left[\begin{array}{cc}A\hfill & M\hfill \\ \hfill & B\hfill \end{array}\right]$, where $A$ and $B$ are Banach algebras (with $U$-module structure) and $M$ is a Banach $A,B$-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949–958], and they showed that the weak module amenability of $2\times 2$ triangular Banach algebra ${{𝒯}}_2$ (as an ${ℐ}:=\left\{\left.\left[\begin{array}{cc}\hfill \alpha \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \alpha \hfill \end{array}\right]\right|\alpha \in U\right\}$-bimodule) is equivalent with the weak module amenability of the corner algebras $A$ and $B$ (as Banach $U$-bimodules). The main aim of this paper is to investigate the module $(\sigma ,\tau )$-amenability and weak module $(\sigma ,\tau )$-amenability of the triangular Banach algebra ${𝒯}$ of order three, where $\sigma$ and $\tau$ are $U$-module morphisms on ${𝒯}$. Also, we give some results for semigroup algebras.

For a finite group $G$, the $k$-characters are class function on subsets of $G^k$ called the $k$-classes of $G$. For a finite group $G$, the $k$-class sums form a basis for a subring of $ℂG^k$ which we call the $k$-$S$-ring of $G$, and we think of these rings as generalized centralizer rings. We show that $2$-characters and extended $2$-characters occur naturally as characters of the $2$-$S$-ring and that $3$-characters occur naturally as characters of the $3$-$S$-ring.

We introduce a notion of generalized Auslander–Reiten duality on a Hom-finite Krull–Schmidt exact category ${𝒞}$. This duality induces the generalized Auslander–Reiten translation functors $\tau$ and ${\tau }^{-}$. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories ${{𝒞}}_r$ and ${{𝒞}}_l$ of ${𝒞}$. A non-projective indecomposable object lies in the domain of $\tau$ if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of ${\tau }^{-}$ if and only if it appears as the first term of an almost split conflation. We study the generalized Auslander–Reiten duality on the category of finitely presented representations of locally finite interval-finite quivers.

We show that a separable equivalence between symmetric algebras preserves the dominant dimensions of certain endomorphism algebras of modules. We apply this to show that the dominant dimension of the category $\mathrm{\text{coMack}}(B)$ of cohomological Mackey functors of a $p$-block $B$ of a finite group with a nontrivial defect group is $2$.

For a nonempty formation $𝔉$, let ${𝔉}^0$ be the least formation containing $𝔉$ such that ${(G\times H)}^{{𝔉}^0}=G^{{𝔉}^0}\times H^{{𝔉}^0}$ for all groups $G$ and $H$. If $𝔉={𝔉}^0$, then $𝔉$ is said to be a DH-formation. A function $f$: $\mathbb{P}\to \{\mathrm{\text{formations}}\}$ is called a formation function and $\mathrm{\text{Supp}}(f)=\{p\in \mathbb{P}:f(p)\ne \varnothing \}$ is called the support of $f$. If $𝔉=\mathrm{\text{SLF}}(f)={𝔈}_{\pi }\bigcap ({\bigcap }_{p\in \pi }{𝔈}_{p^{\prime }}f(p))$ and ${𝔑}_{p}f(p)=f(p)$, where $\pi =\mathrm{\text{supp}}(f)$, then $𝔉$ is said to be a local formation defined by $f$. If ${(f(p))}^0=f(p)$ for all $p\in \mathrm{\text{Supp}}(f)$, then $f$ is called a DH-function.

In this paper, we prove that every local formation can be defined by a DH-function $f$ and each value of the DH-function $f$ is a DH-formation. Besides, we also give the necessary and sufficient condition under which a local Fitting class $𝔉$ is a DH-formation.

Let $𝔞,𝔟$ be ideals of a commutative Noetherian ring $R$ and $N$ be a finitely generated $R$-module. By using filter regular sequences, we show that the infimum of integers $i$ such that the local cohomology modules $H_{𝔞}^i(N)$ and $H_{𝔟}^i(N)$ are not isomorphic is equal to the infimum of the depths of $R_{𝔭}$-modules $N_{𝔭}$, where $𝔭$ runs over all prime ideals of ${Supp}_{R}N$ containing only one of the ideals $𝔞,𝔟$. In particular, these local cohomology modules are isomorphic for all integers $i$ if and only if ${Supp}_R(N/𝔟N)={Supp}_R(N/𝔞N)$. As an application of this result, we prove that for a positive integer $s$, $H_{𝔞}^i(N)$ is Artinian for all $i<s$ if and only if, it can be represented as a finite direct sum of $i\mathrm{\text{th}}$ local cohomology modules of $N$ with respect to some maximal ideals in ${Supp}_R(N/𝔞N)$ for any $i<s$. These representations are unique when they are minimal with respect to inclusion.

In this note, we construct two-sided tilting complexes corresponding to one-sided tilting complexes for Brauer tree algebras.

In this note, we introduce the concept of an almost left automorphic Moufang loop and study the properties of tangent algebras of smooth loops of this class.

Let $R$ be a commutative Noetherian ring and let $I\ne 0$ be a proper ideal of $R$. We study some properties of a family of rings $R{(I)}_{a,b}$ that are obtained as quotients of the Rees algebra associated with the ring $R$ and the ideal $I$. We deal with the strongly cotorsion property of local cohomology modules of $R{(I)}_{a,b}$, when $R{(I)}_{a,b}$ is a local ring. Also, we investigate generically Cohen–Macaulay, generically Gorenstein, and generically quasi-Gorenstein properties of $R{(I)}_{a,b}$. Finally, we show that $R{(I)}_{a,b}$ is approximately Cohen–Macaulay if and only if $R$ is approximately Cohen–Macaulay, provided some special conditions.

The power graph of a group $G$ is the simple graph ${𝒫}(G)$, with vertex-set $G$ and vertices $x$ and $y$ are adjacent, if and only if $x\ne y$ and either $y=x^m$ or $x=y^m$ for some positive integer $m$. The proper power graph of $G$, denoted ${{𝒫}}^{\ast }(G)$, is the graph obtained from ${𝒫}(G)$ by deleting the vertex $1$. In [On the connectivity of proper power graphs of finite groups, Comm. Algebra 43 (2015) 4305–4319], it is proved that if $n\ge 9$ and neither $n$ nor $n-1$ is a prime, then ${{𝒫}}^{\ast }(S_n)$ is connected and $d({{𝒫}}^{\ast }(S_n))\le 26$. In this paper, we improve the diameter bound of ${{𝒫}}^{\ast }(S_n)$ for which ${{𝒫}}^{\ast }(S_n)$ is connected. We show that $d({{𝒫}}^{\ast }(S_9))=d({{𝒫}}^{\ast }(S_{10}))=14$, $d({{𝒫}}^{\ast }(S_{15}))=12$, and $6\le d({{𝒫}}^{\ast }(S_n))\le 10$ for $n\ge 16$. We also describe a number of short paths in these power graphs.

Let $p$ be a prime and let ${ℱ}$ be a saturated fusion system over a finite $p$-group $P$. A fusion system ${ℱ}$ is said to be nilpotent if ${ℱ}={{ℱ}}_P(P)$. We give a criteria for a saturated fusion system ${ℱ}$ to be nilpotent.

Let $H$ be an arbitrary pointed Hopf algebra of rank one and $G$ the group of group-like elements of $H$. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right $H$-comodules into a direct sum of indecomposables. This enables us to describe the corepresentation ring of $H$ in terms of generators and relations. Such a ring is not commutative if $G$ is not abelian. We describe all nilpotent elements of the corepresentation ring of $H$ if $G$ is a finite abelian group or a particular Hamiltonian group. In this case, all nilpotent elements of the corepresentation ring form a principal ideal which is either zero or generated by a nilpotent element of degree 2.

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations $L_c({\lambda }^{\pm })$ of the rational Cherednik algebra $H_c(D_n,{ℂ}^n)$ of type $D$ for symmetric bipartitions $\lambda$ are infinite dimensional for all parameters $c$. In particular, all finite dimensional irreducible representations of rational Cherednik algebras of type $D$ arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type $B$.

In this paper, we present constructions of primitive and non-primitive BCH codes using monoid rings over the local ring ${\mathbb{Z}}_{2^m}$, with $m\ge 1$. We show that there exist two sequences ${\{C_{b^{j}n}\}}_{j\ge 1}$ and ${\{C_{b^{j}n}^{\prime }\}}_{j\ge 1}$ of non-primitive BCH codes (over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_{2^m}$, respectively) against primitive BCH codes $C_n$ of length $n$ and $C_n^{\prime }$ (over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_{2^m}$), respectively. A technique is developed in an innovative way that enables the data path to shift instantaneously during transmission via the coding schemes of $C_n$, $C_n^{\prime }$, ${\{C_{b^{j}n}\}}_{j\ge 1}$ and ${\{C_{b^{j}n}^{\prime }\}}_{j\ge 1}$. The selection of the schemes is subject to the choice of better code rate or better error-correction capability of the code. Finally, we present a decoding procedure for BCH codes over Galois rings, which is also used for the decoding of BCH codes over Galois fields, based on the modified Berlekamp–Massey algorithm.

Let $R,T$ be commutative rings with identity such that $R\subseteq T$. A ring extension $R\subseteq T$ is called a $\mathrm{\Delta }$-extension of rings if $R_1+R_2$ is a subring of $T$ for each pair of subrings $R_1,R_2$ of $T$ containing $R$. In this paper, a characterization of integrally closed $\mathrm{\Delta }$-extension of rings is given. The equivalence of $\mathrm{\Delta }$-extension of rings and $\lambda$-extension of rings is established for an integrally closed extension of a local ring. Over a finite dimensional, integrally closed extension of local rings, the equivalence of $\mathrm{\Delta }$-extensions of rings, FIP, and FCP is shown. Let $R$ be a subring of $T$ such that $R$ is invariant under action by $G$, where $G$ is a subgroup of the automorphism group of $T$. If $R\subseteq T$ is a $\mathrm{\Delta }$-extension of rings, then $R^G\subseteq T^G$ is a $\mathrm{\Delta }$-extension of rings under some conditions. Many such $G$-invariant properties are also discussed.

A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.