Journal of Algebra and Its Applications

In this paper, we introduce and study the notion of quasi-multipliers on a semi-topological semigroup $S$. The set of all quasi-multipliers on $S$ is denoted by $𝔔𝔐(S)$. First, we study the problem of extension of quasi-multipliers on topological semigroups to its Stone–Čech compactification. Indeed, we prove if $S$ is a topological semigroup such that $S\times S$ is pseudocompact, then $𝔔𝔐(S)$ can be regarded as a subset of $𝔔𝔐(\beta S).$ Moreover, with an extra condition we describe $𝔔𝔐(S)$ as a quotient subsemigroup of $\beta S.$ Finally, we investigate quasi-multipliers on topological semigroups, its relationship with multipliers and give some concrete examples.

We study some properties of formal local cohomology modules ${{ℱ}}_a^i(M)$ in Serre subcategories. As a consequence, we obtain some results on the minimax modules. We also describe the sets ${\mathrm{Cos}}_R({{ℱ}}_a^i(M))$ and ${\mathrm{Coass}}_R({{ℱ}}_a^i(M))$.

We introduce the notion of semi-poor modules and consider the possibility that all modules are either injective or semi-poor. This notion gives a generalization of poor modules that have minimal injectivity domain. A module $M$ is called semi-poor if whenever it is $N$-injective and $N\ne 0$, then the module $N$ has nonzero socle. In this paper the properties of semi-poor modules are investigated and are used to characterize various families of rings. We introduce the rings over which every module is either semi-poor or injective and call such condition property $(P)$. The structure of the rings that have the property $(P)$ is completely determined. Also, we give some characterizations of rings with the property $(P)$ in the language of the lattice of hereditary pretorsion classes over a given ring. It is proved that a ring $R$ has the property $(P)$ iff either $R$ is right semi-Artinian or $R=R_1\times R_2$ where $R_1$ is a semisimple Artinian ring and $R_2$ is right strongly prime and a right $c$-ring with zero right socle.

We construct an example of an IC ring where perspectivity is transitive, but not all isomorphic idempotents are perspective. We also develop new criteria for checking perspectivity of idempotents in rings.

Let $R$ be a commutative Noetherian ring. For a finitely generated $R$-module $M$, Northcott introduced the reducibility index of $M$, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ in $M$. On the other hand, for an Artinian $R$-module $A$, Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of $A$ does not depend on the choice of the representation. This number is called the sum-reducibility index of $A$. In the former part of this paper, we compute the reducibility index of $S{\otimes }_{R}M$, where $R\to S$ is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of $R$ are studied. For the latter part of this paper, we clarify the relation among the reducibility index of $M$, that of the completion of $M$, and the sum-reducibility index of the Matlis dual of $M$.

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring along an ideal. Our results generate examples that enrich the current literature with new and original families of rings satisfying these properties.

A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang–Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions obtained by Rump in A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193 (2005) 40–55, https://doi.org/10.1016/j.aim.2004.03.019 and answer positively a question posed by Cedó et al. in Question 4.2 in Structure monoids of set-theoretic solutions of the Yang–Baxter equation, preprint (2019), https://arxiv.org/abs/1912.09710. Moreover, we develop a theory of extensions for left non-degenerate set-theoretic solutions of the Yang–Baxter equation that allows one to construct new families of set-theoretic solutions.

The Bochner–Schoenberg–Eberlein (BSE)-property on commutative Banach algebras is a property related to multiplier algebras of Banach algebras. In this paper, we answer the problem (12) raised by Javanshiri and Nemati in [Amalgamated duplication of the Banach algebra $𝔄$ along a $𝔄$-bimodule ${𝒜}$, J. Algebra Appl. 17(9) (2018) 1850169-1–1850169-21]. In this paper, under certain conditions, we show that the amalgamated Banach algebra ${𝒜}⋊𝔄$ is BSE-algebra if and only if ${𝒜}$ and $𝔄$ are BSE-algebras.

Let $G$ be a finite group, $p$ a prime, $P$ a Sylow $p$-subgroup of $G$ and $d$ a power of $p$ such that $1<d<\left|P\right|$. Let $G_p^{\ast }$ denote the unique smallest normal subgroup of $G$ for which the corresponding factor group is abelian of exponent dividing $p-1$. Let ${𝔉}_1$, ${𝔉}_2$, ${𝔉}_3$ be classes of all $p$-groups, $p$-nilpotent groups and $p$-supersolvable groups, respectively, $G^{𝔉}$ be the $𝔉$-residual of $G$. Let $X\in \{{(G_p^{\ast })}^{{𝔉}_1},G^{{𝔉}_2},G^{{𝔉}_3}\}$. A subgroup $H$ of a finite group $G$ is said to have $\mathrm{\Pi }$-property in $G$, if for any $G$-chief factor $L/K$, $|G/K:N_{G/K}((H\cap L)K/K)|$ is a $\pi ((H\cap L)K/K)$-number. A normal subgroup $E$ of $G$ is said to be $p$-hypercyclically embedded in $G$ if every $p$-$G$-chief factor of $E$ is cyclic, where $p$ is a fixed prime. In this paper, we prove that $E$ is $p$-hypercyclically embedded in $G$ if and only if for some $p$-subgroups $H$ of $E$, $H\cap X$ have $\mathrm{\Pi }$-property in $G$.

In this paper, we prove that if $R$ is a local ring of dimension $d,$ $d\ge 2$ and $\frac{1}{d!}\in R$ then the group $\frac{{\mathrm{\text{Um}}}_{d+1}(R[X])}{E_{d+1}(R[X])}$ has no $k$-torsion, provided $\mathrm{\text{kR}}=R.$ We also prove that if $R$ is a regular ring of dimension $d,$ $d\ge 2$ and $\frac{1}{d!}\in R$ such that $E_{d+1}(R)$ acts transitively on ${\mathrm{\text{Um}}}_{d+1}(R)$ then $E_{d+1}(R[X])$ acts transitively on ${\mathrm{\text{Um}}}_{d+1}(R[X]).$

In this paper, we study the behavior of endomorphism rings of indecomposable (uniform) quasi-injective modules. A very natural question here is, for a morphism $f:A\to B$, with $A,B$ indecomposable (uniform) quasi-injective right $R$-modules, and $g:E(A)\to E(B)$ an extension of $f$ where $E(-)$ denotes the injective hull, what is the relation between kernels of $f$ and $g$, their monogeny classes and their upper parts?

Rings in which each finitely generated right ideal is automorphism-invariant (right$fa$-rings) are shown to be isomorphic to a formal matrix ring. Among other results it is also shown that (i) if $R$ is a right nonsingular ring and $n>1$ is an integer, then $R$ is a right self injective regular ring if and only if the matrix ring ${𝕄}_n(R)$ is a right $fa$-ring, if and only if ${𝕄}_n(R)$ is a right automorphism-invariant ring and (ii) a right nonsingular ring $R$ is a right $fa$-ring if and only if $R$ is a direct sum of a square-full von Neumann regular right self-injective ring and a strongly regular ring containing all invertible elements of its right maximal ring of fractions. In particular, we show that a right semiartinian (or left semiartinian) ring $R$ is a right nonsingular right $fa$-ring if and only if $R$ is a left nonsingular left $fa$-ring.

In 2004, Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a $\mathbb{Z}$-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that cannot be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.

Let $D$ be an integral domain with quotient field $K,$ throughout$.$ Call two elements $x,y\in D\setminus \{0\}$$v$-coprime if $xD\cap yD=xyD.$ Call a nonzero non-unit $r$ of an integral domain $D$ rigid if for all $x,y|r$ we have $x|y$ or $y|x.$ Also, call $D$ semirigid if every nonzero non-unit of $D$ is expressible as a finite product of rigid elements. We show that a semirigid domain $D$ is a GCD domain if and only if $D$ satisfies $\ast :$ product of every pair of non-$v$-coprime rigid elements is again rigid. Next, call $a\in D$ a valuation element if $aV\cap D=aD$ for some valuation ring $V$ with $D\subseteq V\subseteq K$ and call $D$ a VFD if every nonzero non-unit of $D$ is a finite product of valuation elements. It turns out that a valuation element is what we call a packed element: a rigid element $r$ all of whose powers are rigid and $\sqrt{rD}$ is a prime ideal. Calling $D$ a semi-packed domain (SPD) if every nonzero non-unit of $D$ is a finite product of packed elements, we study SPDs and explore situations in which a variant of an SPD is a semirigid GCD domain.

We study canonical bases of a subalgebra $A=𝕂[[f_1,\dots ,f_s]]\subseteq 𝕂[[x_1,\dots ,x_n]]$ over a field $𝕂$, and we associate with $A$ a fan called the canonical fan of $A$. This generalizes the notion of the standard fan of an ideal.

In this paper, we characterize the (covariant) isotropy groups of free, finitely generated racks and quandles. As a consequence, we show that the usual inner automorphisms of such racks and quandles are precisely those automorphisms that are “coherently extendible”. We then use this result to compute the global isotropy groups of the categories of racks and quandles, i.e. the automorphism groups of the identity functors of these categories.

Let $R$ be a commutative ring with identity, $X$ be an indeterminate and $I_c(R)$ be the set of elements $a$ of $R$ such that there exists an $R$-homomorphism of rings $\sigma :R[[X]]\to R$ with $\sigma (X)=a$. O’Malley called $R$ to be power invariant (respectively, strongly power invariant) if whenever $S$ is a ring such that $R[[X]]$ is isomorphic to $S[[X]]$ (respectively, whenever $S$ is a ring and $\phi$ is an isomorphism of $R[[X]]$ onto $S[[X]]$), then $R$ and $S$ are isomorphic (respectively, then there exists an $S$-automorphism $\psi$ of $S[[X]]$ such that $\psi (X)=\phi (X)$) [M. O’Malley, Isomorphic power series rings, Pacific J. Math. 41(2) (1972) 503–512]. We prove that a ring $R$ is power invariant in each of the following case: $(1)$ $R$ is a domain in which $I_c(R)$ is comparable to each radical ideal of $R$ (for instance a domain with Krull dimension one), $(2)$ $R$ is a domain in which Jac$(R)$ (i.e. the Jacobson radical of $R$) is comparable to each radical ideal of $R$ and $(3)$ $R$ is a Prüfer domain. Also in each of the aforementioned case, we prove that either $R$ is strongly power invariant or $R$ is isomorphic to a quasi-local power series ring. Let $M$ be a unital module over $R$. We show that if $R$ is reduced and strongly power invariant, then Nagata’s idealization ring $R(+)M$ is strongly power invariant (but the converse is false). Ishibashi called a ring $R$ to be strongly$n$-power invariant if whenever $S$ is a ring and $\phi$ is an isomorphism of $R[[X_1,\dots ,X_n]]$ onto $S[[X_1,\dots ,X_n]]$, then there exists an $S$-automorphism $\psi$ of $S[[X_1,\dots ,X_n]]$ such that $\psi (X_i)=\phi (X_i)$ for each $i$. We prove that if $R$ is a ring in which $I_c(R)$ is nil, then $R$ is strongly$n$-power invariant for all positive integer $n$. We deduce that every polynomial ring in finitely many indeterminates is strongly$n$-power invariant for all positive integer $n$.

Let ${𝒰}$ be a 2-torsion free unital generalized matrix algebra, and $\varphi :{𝒰}\to {𝒰}$ be a linear map satisfying

In this paper, we mainly investigate the class-preserving Coleman automorphisms of finite groups whose Sylow $2$-subgroups are semidihedral. We prove that if $G$ is a finite solvable group with semidihedral Sylow $2$-subgroups, then ${\mathrm{\text{Out}}}_c(G)\cap {\mathrm{\text{Out}}}_{\mathrm{\text{Col}}}(G)$ is a $2^{\prime }$-group and therefore $G$ satisfies the normalizer property. As some applications of this result, we also investigate the normalizer property of the following groups: the groups whose Sylow $2$-subgroups are semidihedral and Sylow subgroups of odd order are all cyclic, the groups $G=N⋊S$ with $N$ a nilpotent normal subgroup and $S$ a maximal class $2$-group, and the wreath products $G=H\wr Q$ with $H$ a group whose Sylow $2$-subgroups are of maximal class with order $\ge 8$ and $Q$ a rational permutation group.

Let $R$ be a ring. We study the FG-pure exact sequences, which are defined analogously to pure exact sequences except with respect to finitely generated $R$-modules rather than the finitely presented $R$-modules. For Noetherian rings the two notions coincide. But for non-Noetherian rings, the FG-purity is sharper than the usual purity. We also study the corresponding notion of FG-flat module and construct a model for the FG-pure derived category. It is a compactly generated triangulated category.

Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a division ring of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs).

As an application of our methods, we prove that crossed products of division rings with free-by-{infinite cyclic} and surface groups are pseudo-Sylvester domains unconditionally and Sylvester domains if and only if they admit stably free cancellation. This relies on the recent proof of the Farrell–Jones conjecture for normally poly-free groups and extends previous results of Linnell–Lück and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products.