Journal of Algebra and Its Applications

Let $(R,𝔪)$ be a Noetherian local ring and $A$ be an Artinian $R$-module. Consider the following property for $A$ :

Let $R$ be a commutative Noetherian ring, $I$ and $J$ be two ideals of $R$ and $M$ be an $R$-module (not necessary $I$-torsion). In this paper among other things, it is shown that if dim $M\le 1$, then the $R$-module ${\mathrm{\text{Ext}}}_R^i(R/I,M)$ is finitely generated, for all $i\ge 0$, if and only if the $R$-module ${\mathrm{\text{Ext}}}_R^i(R/I,M)$ is finitely generated, for $i=0,1$. As a consequence, we prove that if $M$ is finitely generated and $t\in \mathbb{N}$ such that the $R$-module ${\mathrm{\text{H}}}_{I,J}^i(M)$ is ${\mathrm{\text{FD}}}_{\le 1}$ (or weakly Laskerian) for all $i<t$, then ${\mathrm{\text{H}}}_{I,J}^i(M)$ is $(I,J)$-cofinite for all $i<t$ and for any ${\mathrm{\text{FD}}}_{\le 0}$ (or minimax) submodule $N$ of ${\mathrm{\text{H}}}_{I,J}^t(M)$, the $R$-modules ${\mathrm{\text{Hom}}}_R(R/I,{\mathrm{\text{H}}}_{I,J}^t(M)/N)$ and ${\mathrm{\text{Ext}}}_R^1(R/I,{\mathrm{\text{H}}}_{I,J}^t(M)/N)$ are finitely generated. Also it is shown that if dim $M/𝔞M\le 1$ (e.g. dim $R/𝔞\le 1$) for all $𝔞\in \tilde{W}(I,J)$, then the local cohomology module ${\mathrm{\text{H}}}_{I,J}^i(M)$ is $(I,J)$-cofinite for all $i\ge 0$.

The classical Kronecker function ring construction associates to a domain $R$ a Bézout domain. Let $R$ be a subring of a ring $S$, and let ⋆ be a star operation on the extension $R\subseteq S$. In their book [Manis Valuations and Prüfer Extensions II, Lectures Notes in Mathematics, Vol. 2103 (Springer, Cham, 2014)], Knebusch and Kaiser develop a more general construction of the Kronecker function ring of $R\subseteq S$ with respect to ⋆. We characterize in several ways, under relatively mild assumption on $R\subseteq S$, the Kronecker function ring as defined by Knebusch and Kaiser. In particular, we focus on the case where $R\subseteq S$ is a flat epimorphic extension or a Prüfer extension.

Based on the concepts of composition ring and composition hyperring, in this note, we introduce the notion of $n$-ary composition hyperring and study the connections with composition hyperrings. Moreover, we show that special strong multiendomorphisms of hyperrings can determine the $n$-ary composition structure of a such hyperring and also a subclass of $n$-ary composition hyperrings. Finally, the three isomorphism theorems are studied in the framework of $n$-ary composition hyperrings.

Let $A$ and $B$ be two rings, $J$ an ideal of $B$ and $f:A\to B$ be a ring homomorphism. The ring $A{\bowtie }^{f}J:=\{(a,f(a)+j)|a\in A,\mathrm{\text{and}}j\in J\}$ is called the amalgamation of $A$ with $B$ along $J$ with respect to $f$. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which $A{\bowtie }^{f}J$, and some related constructions, is either a Hilbert ring, a $G$-domain or a $G$-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the $G$-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

We prove that an isomorphism between saturated fusion systems over the same finite $p$-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if $p$ is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at most $4$ if $p=2$. For odd $p$, this has implications for mod $p$ group cohomology.

Let $(A,〈\cdot ,\dots ,\cdot 〉)$ be an $n$-algebra of arbitrary dimension and over an arbitrary base field $𝔽$. A basis ${ℬ}={\{e_i\}}_{i\in I}$ of $A$ is said to be multiplicative if for any $i_1,\dots ,i_n\in I$, we have either $〈e_{i_1},\dots ,e_{i_n}〉=0$ or $0\ne 〈e_{i_1},\dots ,e_{i_n}〉\in 𝔽e_j$ for some (unique) $j\in I$. If $n=2$, we are dealing with algebras admitting a multiplicative basis while if $n=3$ we are speaking about triple systems with multiplicative bases. We show that if $A$ admits a multiplicative basis then it decomposes as the orthogonal direct sum $A={\oplus }_{\alpha }{I}_{\alpha },$ of well-described ideals admitting each one a multiplicative basis. Also, the minimality of $A$ is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.

We characterize finite $p$-groups of order up to $p^7$ whose central automorphisms fixing the center element-wise are all inner.

We introduce some sequences of binary constant weight codes which are constructed from the affine constructions of balanced incomplete block designs. They give the values for the function $A(n,d,w)$ of constant weight codes, and they are optimal since those values reach the Johnson Bound. We focus on the values obtained by this method.

We classify Borel–Schur algebras having finite representation type. We also determine Auslander–Reiten sequences for a large class of simple modules over Borel–Schur algebras. A partial information on the structure of the socles of Borel-Schur algebras is given.

We say a ring $R$ is $\pi$-Baer if the right annihilator of every projection invariant left ideal of $R$ is generated by an idempotent element of $R$. In this paper, we study connections between the $\pi$-Baer condition and related conditions such as the Baer, quasi-Baer and $\pi$-extending conditions. The $2$-by-$2$ generalized triangular and the $n$-by-$n$ triangular $\pi$-Baer matrix rings are characterized. Also, we prove that a $n$-by-$n$ full matrix ring over a $\pi$-Baer ring is a $\pi$-Baer ring. In contrast to the Baer condition, it is shown that the $\pi$-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.

The notion of context-equivalent algebras introduced by Müller generalizes Morita-equivalence. It is a coarser equivalence in the class of algebras but it still preserves many ring-theoretic properties. It also gives a new equivalence in the class of commutative algebras. We present a version of this equivalence which includes nonunital algebras. In particular, it allows one to relate properties of an algebra and its ideals. We give a criterion for context-equivalent algebras in terms of finite-rank and adjointable operators. Further, we find a better criterion for equivalence for commutative algebras in terms of ideals.

Let $\sigma =\{{\sigma }_i|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a ${\sigma }_i$-group for some $i=i(H/K)$. A set ${\mathscr{H}}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of ${\mathscr{H}}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i\in \sigma$ and ${\mathscr{H}}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every $i$ such that ${\sigma }_i\cap \pi (G)\ne \varnothing$. A subgroup $A$ of $G$ is said to be $\sigma$-quasinormal or $\sigma$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set ${\mathscr{H}}$ such that $A{H}^x=H^{x}A$ for all $x\in G$ and all $H\in {\mathscr{H}}$. We obtain a new characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-permutability is a transitive relation in $G$.

The conjugacy relation plays an important role in group theory. If $a$ and $b$ are elements of a group $G$, $a$ is conjugate to $b$ if $g^{-1}ag=b$ for some $g\in G$. The group conjugacy extends to inverse semigroups in a natural way: for $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$ if $g^{-1}ag=b$ and $gb{g}^{-1}=a$ for some $g\in S^1$. In this paper, we define a conjugacy for an arbitrary semigroup $S$ that reduces to the inverse semigroup conjugacy if $S$ is an inverse semigroup. (None of the existing notions of conjugacy for semigroups has this property.) We compare our new notion of conjugacy with existing definitions, characterize the conjugacy in basic transformation semigroups and their ideals using the representation of transformations as directed graphs, and determine the number of conjugacy classes in these semigroups.

Given a lattice $𝕃$ and a class $K$ of algebraic structures, we say that $𝕃$ forces nilpotency in $K$ if every algebra $A\in K$ whose congruence lattice $\mathrm{\text{Con}}(A)$ is isomorphic to $𝕃$ is nilpotent. We describe congruence lattices that force nilpotency, supernilpotency or solvability for some classes of algebras. For this purpose, we investigate which commutator operations can exist on a given congruence lattice.

This paper shows that, for any positive integer $s$, there exist $s$-arc-transitive digraphs which have non-isomorphic in-local action and out-local action.

In this paper, first, we give a Gröbner–Shirshov pair of finite-dimensional irreducible module $V_q(\lambda )$ over $U_q(A_n),$ the quantized enveloping algebra of type $A_n,$ by using the double free module method and the known Gröbner–Shirshov basis of $U_q(A_n).$ Then, by specializing a suitable version of $U_q(A_n)$ at $q=1,$ we get a Gröbner–Shirshov basis of $U(A_n),$ and get a Gröbner–Shirshov pair for the finite-dimensional irreducible module $V(\lambda )$ over $U(A_n)$.

For $n\in \mathbb{N}$, let ${𝒮}in{g}_n$ be the semigroup of all singular mappings on $X_n=\{1,2,\dots ,n\}$. For each nonempty subset $A$ of $X_n$, let ${{𝒮}}_n(A)=\{\alpha \in {𝒮}in{g}_n:(\forall k\in A)x\le k\Rightarrow x\alpha \le k\}$ be the semigroup of all $A$-decreasing mappings on $X_n$. In this paper we determine the rank and idempotent rank of the semigroup ${{𝒮}}_n(A)$.

For an (oriented) graph $G$ on the vertex set $\{0,1,\dots ,n\}$ (rooted at $0$), Postnikov and Shapiro (Trans. Amer. Math. Soc. 356 (2004) 3109–3142) associated a monomial ideal ${{ℳ}}_G$ in the polynomial ring $R=k[x_1,\dots ,x_n]$ over a field $k$ such that the number of standard monomials of $\frac{R}{{{ℳ}}_G}$ equals the number of (oriented) spanning trees of $G$ and hence, ${dim}_k(\frac{R}{{{ℳ}}_G})=det(L_G)$, where $L_G$ is the truncated Laplace matrix of $G$. The standard monomials of $\frac{R}{{{ℳ}}_G}$ correspond bijectively to the $G$-parking functions. In this paper, we study a monomial ideal $J_n$ in $R$ having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal $J_n$ is the cellular resolution supported on a subcomplex of the first barycentric subdivision $Bd({\mathrm{\Delta }}_{n-1})$ of an $n-1$ simplex ${\mathrm{\Delta }}_{n-1}$. The integer sequence ${\{{dim}_k(\frac{R}{J_n})\}}_{n\ge 1}$ has many interesting properties. In particular, we obtain a formula, ${dim}_k(\frac{R}{J_n})=det({[m_{ij}]}_{n\times n})$, with $m_{ij}=1$ for $i>j$, $m_{ii}=i$ and $m_{ij}=i-j$ for $i<j$, similar to ${dim}_k(\frac{R}{{{ℳ}}_G})=det(L_G)$.

In this paper, we shall describe costandard modules $\nabla (\lambda )$ of restricted highest weight $\lambda$ for Schur superalgebra $S(2|2)$ over an algebraically closed field $K$ of positive characteristic $p>2$. Additionally, for a restricted highest weight $\lambda$, we determine all composition factors of the costandard module $\nabla (\lambda )$; in particular we compute the decomposition numbers in the process of the modular reduction of a simple module with a restricted highest weight.