Journal of Algebra and Its Applications

A subgroup $H$ of a group $G$ is said to be an $\mathcal{\mathscr{H}}C$-subgroup of $G$, if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H^g\cap N_K(H)\le H$, for all $g\in G$. In this paper, we investigate the structure of groups based on the assumption that every subgroup of $P\cap G^{{{𝒩}}_p}$ of order $p$ or 4 (if $p=2$) is an $\mathcal{\mathscr{H}}C$-subgroup of $N_G(P)$, here $P$ is a Sylow $p$-subgroup of $G$. Some results for a group to be $p$-nilpotent and supersolvable are obtained and many known results are generalized.

A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $\left|B\right|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\mathrm{\Delta }[G]$. We prove that for every $n\in \mathbb{N}$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4\left|n\right|-3}}{2}\le \mathrm{\Delta }[C_n]\le \frac{3}{2}\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$), then $\mathrm{\Delta }[C_n]\le \frac{12}{\sqrt{73}}\sqrt{n}$ (and $\mathrm{\Delta }[C_n]<\frac{2}{\sqrt{3}}\sqrt{n}$). Also, we calculate the difference sizes of all cyclic groups of cardinality $\le 100$.

In this paper, we define an equivalence relation induced by $n$-ary subpolygroups and show that such relation is full conjugation when $n$-ary subpolygroup is normal. Then, we introduce a construction and some properties of direct product of $n$-ary polygroups via $n$-ary factor polygroups. This construction in principle is enlargement of elements of $n$-ary polygroup except zero. Finally, we introduce and investigate the semidirect product of $n$-ary polygroups via $n$-ary factor polygroups.

We determine a supercharacter theory for the matrix Sylow $p$-subgroup ${}^3{D}_4^{\mathrm{\text{syl}}}(q^3)$ of the Steinberg triality group ${}^3{D}_4(q^3)$, and establish the supercharacter table of ${}^3{D}_4^{\mathrm{\text{syl}}}(q^3)$.

In analogy to the fact that isomorphic idempotents arise from flipped factorizations, we show that under some weak conditions, partial factorizations of the complement idempotents may also be preserved. Applying these results, we give new characterizations of exchange rings. We also show that the definition of an exchange element is left–right symmetric.

Let $(W,S,L)$ be a weighted Coxeter group such that the order $m_{st}$ of the product $st$ is not 3 for any $s,t\in S$ and that $N(W):=max\{L(w_I)|I\subseteq S;\left|W_I\right|<\infty \}<\infty$, where $w_I$ is the longest element in the parabolic subgroup $W_I$ of $W$ generated by $I$. We prove that $(W,S,L)$ is bounded with $N(W)$ an upper bound in the sense of Lusztig in Sec. 13.2 of [Hecke Algebras with Unequal Parameters, arXiv:math/0208154 v2 [math.RT] 10 Jun 2014], verifying a conjecture of Lusztig in our case (see Conjecture 13.4 in loc. cite).

We give the necessary and sufficient condition for a separated Cohn–Leavitt path algebra of a finite digraph to have Invariant Basis Number (IBN). As a consequence, separated Cohn path algebras have IBN. We determine the non-stable K-theory of a corner ring in terms of the non-stable K-theory of the ambient ring. We give a necessary condition for a corner algebra of a separated Cohn–Leavitt path algebra of a finite graph to have IBN. We provide Morita equivalent rings which are non-IBN, but are of different types.

The subgroup structure of a finite group, under the assumption that its every non-nilpotent maximal subgroup has prime index, is studied in the paper.

Let $G$ be a group and $\mathrm{\text{Inn}}(G)$, ${\mathrm{\text{Aut}}}_{pwi}(G)$, ${\mathrm{\text{Aut}}}_c(G)$ and $D(G)$ denote the group of all inner automorphisms, the group of all pointwise inner automorphisms, the group of all central automorphisms and the group of all derival automorphisms of $G$, respectively. We know that in a finite $p$-group $G$ of class 2, $D(G)=\mathrm{\text{Inn}}(G)$ if and only if $G^{\prime }$ is cyclic and $D(G)=D^{\ast }(G)$, where $D^{\ast }(G)$ is the group of all derival automorphisms of $G$ which fix $Z(G)$ elementwise. In this paper, we characterize all finite nilpotent groups of class 2 for which $D(G)$ or $D^{\ast }(G)$ is equal to $\mathrm{\text{Inn}}(G)$, $C^{\ast }(G)$, ${\mathrm{\text{Aut}}}_c(G)$ and ${\mathrm{\text{Aut}}}_{pwi}(G)$. Also, we characterize all finitely generated nilpotent groups of class 2 for which $D^{\ast }(G)$ is equal to $\mathrm{\text{Inn}}(G)$ and give some interesting corollaries in this regard.

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an $R$-module, where $R$ is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring $R$ with non-essential Noetherian dimension is uniform.

Let $R$ be a commutative Noetherian ring, $\mathrm{\Phi }$ a system of ideals of $R$ and $I\in \mathrm{\Phi }$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\mathrm{\text{dim}}M\le 1$, then the $R$-module ${\mathrm{\text{Ext}}}_R^i(R/I,M)$ is weakly Laskerian, for all $i\ge 0$, if and only if the $R$-module ${\mathrm{\text{Ext}}}_R^i(R/I,M)$ is weakly Laskerian for $i=0,1$. Let $t\in {\mathbb{N}}_0$ be an integer and $M$ an $R$-module such that ${\mathrm{\text{Ext}}}_R^i(R/I,M)$ is weakly Laskerian for all $i\le t+1$. We prove that if the $R$-module ${\mathrm{\text{H}}}_{\mathrm{\Phi }}^i(M)$ is ${\mathrm{\text{FD}}}_{\le 1}$ for all $i<t$, then ${\mathrm{\text{H}}}_{\mathrm{\Phi }}^i(M)$ is $\mathrm{\Phi }$-weakly cofinite for all $i<t$, and for any ${\mathrm{\text{FD}}}_{\le 0}$ (or minimax) submodule $N$ of ${\mathrm{\text{H}}}_{\mathrm{\Phi }}^t(M)$, the $R$-modules ${\mathrm{\text{Hom}}}_R(R/I,{\mathrm{\text{H}}}_{\mathrm{\Phi }}^t(M)/N)$ and ${\mathrm{\text{Ext}}}_R^1(R/I,{\mathrm{\text{H}}}_{\mathrm{\Phi }}^t(M)/N)$ are weakly Laskerian. Let $N$ be a finitely generated $R$-module. We also prove that ${\mathrm{\text{Ext}}}_R^j(N,{\mathrm{\text{H}}}_{\mathrm{\Phi }}^i(M))$ and ${\mathrm{\text{Tor}}}_j^R(N,H_{\mathrm{\Phi }}^i(M))$ are $\mathrm{\Phi }$-weakly cofinite for all $i$ and $j$ whenever $M$ is weakly Laskerian and ${\mathrm{\text{H}}}_{\mathrm{\Phi }}^i(M)$ is ${\mathrm{\text{FD}}}_{\le 1}$ for all $i$. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.

Let $G$ be a solvable, nonnilpotent and directly indecomposable finite group. Suppose $G$ has exactly two nonlinear nonfaithful irreducible characters ${\chi }_1,{\chi }_2$. In this paper, we classify the finite groups that satisfy these properties with $\mathrm{\text{Ker}}{\chi }_1\cap \mathrm{\text{Ker}}{\chi }_2>1$.

A high cited paper [Feng and Li, One-regular graphs of square-free order of prime valency, Europ. J. Combin. 32 (2011) 261–275] classified 1-arc-regular prime-valent graphs (necessarily Cayley graphs) of square-free order. It is thus interesting to ask what are the $s$-arc-regular prime-valent Cayley graphs of square-free order for each $s\ge 2$. In this paper, a complete list of all such graphs is given. The proving process also involves a complete list of the vertex stabilizers of $s$-arc-regular prime-valent graphs of any order.

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers $n$ and $t$, we consider the monomial ideal $I={\mathrm{\text{Ind}}}_t(P_n)$ generated by all monomials $x^F$, where $F$ is an independent set of vertices of the path graph $P_n$ of size $t$, which is indeed the facet ideal of the $t$th skeleton of the independence complex of $P_n$. We describe the set of associated primes of all powers of $I$ explicitly. It turns out that any such ideal $I$ has the persistence property. Moreover, the index of stability of $I$ and the stable set of associated prime ideals of $I$ are determined.

Let $P$ be a finite abelian $p$-group and $𝔽$ a finite field of characteristic $p$. We give an algorithm to write down a set of generators of $K_2(𝔽[P])$ via a simple presentation of it and obtain some formulae for counting these generators. Using Witt decomposition, we also determine the group structure of $K_3(𝔽[C_{p^n}])$.

Suppose that $X_A\subset {\mathbb{P}}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper, we compute the Euclidean distance degree and polar degrees of $X_A$ (along with other associated invariants) combinatorially working from the matrix $B$. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix $A$ in the codimension two case.

A ring $R$ is called eversible if every left zero-divisor in $R$ is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that $R$ is eversible if and only if its upper triangular matrix ring $T_n(R)$ is eversible, and if $M_n(R)$ is eversible then $R$ is eversible.

Let $R$ be a Cohen–Macaulay local ring. We prove that the $n$th syzygy module of a maximal Cohen–Macaulay $R$-module cannot have a semidualizing direct summand for every $n\ge 1$. In particular, it follows that $R$ is Gorenstein if and only if some syzygy of a canonical module of $R$ has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

Let $\mathcal{𝒞}$ be a category with an involution ∗. Suppose that $\phi :X\to X$ is a morphism and $({\phi }_1,Z,{\phi }_2)$ is an (epic, monic) factorization of $\phi$ through $Z$, then $\phi$ is core invertible if and only if ${({\phi }^{\ast })}^2{\phi }_1$ and ${\phi }_2{\phi }_1$ are both left invertible if and only if $({({\phi }^{\ast })}^2{\phi }_1,Z,{\phi }_2)$, $({\phi }_2^{\ast },Z,{\phi }_1^{\ast }{\phi }^{\ast }\phi )$ and $({\phi }^{\ast }{\phi }_2^{\ast },Z,{\phi }_1^{\ast }\phi )$ are all essentially unique (epic, monic) factorizations of ${({\phi }^{\ast })}^2\phi$ through $Z$. We also give the corresponding result about dual core inverse. In addition, we give some characterizations about the coexistence of core inverse and dual core inverse of an $R$-morphism in the category of $R$-modules of a given ring $R$.

Let $A$ be a principal ideal domain (PID) or more generally a Dedekind domain and let $F$ be a coherent functor from the category of finitely generated $A$-modules to itself. We classify the half-exact coherent functors $F$. In particular, we show that if $F$ is a half-exact coherent functor over a Dedekind domain $A$, then $F$ is a direct sum of functors of the form ${\mathrm{\text{Hom}}}_A(P,-)$, ${\mathrm{\text{Hom}}}_A(A/{𝔭}^s,-)$ and $A/{𝔭}^s\otimes -$, where $P$ is a finitely generated projective $A$-module, $𝔭$ a nonzero prime ideal in $A$ and $s\ge 1$.

The notion of an Ohm–Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan–Hochster proof of Stillman’s conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from [N. Epstein and J. Shapiro, The Ohm-Rush content function, J. Algebra Appl. 15(1) (2016) 1650009, 14 pp.] of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Prüfer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions.

Let $A$ be a finite-dimensional $k$-algebra. In this paper, we mainly study the representation type of subcategories of the bounded derived category $D^b(A)$. First, we define the representation type and some homological invariants including cohomological length, width, range for subcategories. In this framework, we provide a characterization for derived discrete algebras. Moreover, for a finite-dimensional algebra $A$, we establish the first Brauer–Thrall type theorem of certain contravariantly finite subcategories ${𝒞}$ of $D^b(A)$, that is, ${𝒞}$ is of finite type if and only if its cohomological range is finite.