Journal of Algebra and Its Applications

Let $R$ be an integral domain with only finitely many overrings, equivalently, a domain such that its integral closure $R^{\prime }$ is a Prüfer domain with finite spectrum and there are only finitely many rings between $R$ and $R^{\prime }$. Jaballah solved the problem of counting the overrings in the case $R=R^{\prime }$ but left the general case as an open problem [A. Jaballah, The number of overrings of an integrally closed domain, Expo. Math. 23 (2005) 353–360, Problem 3.4]. The purpose of this paper is to provide a solution to that problem.

We study completely simple congruences on an arbitrary $E$-inversive semigroup $S$. In particular, we show that every such congruence $\rho$ on $S$ is uniquely determined by its kernel and trace, and that the trace of $\rho$ is a congruence on the biordered set $E_S$. Moreover, we investigate the complete lattice of all completely simple congruences on $S$ and show that the trace relation is a complete congruence on this lattice. We also construct a family of completely simple congruences on $S$.

Let $P$ be a Sylow $p$-subgroup and $b(G)$ be the largest irreducible character degree of a finite nonabelian group $G$. Then $|P/O_p(G)|\le {(b{(G)}^p/p)}^{\frac{1}{p-1}}$.

For a Hopf von Neumann algebra $(\mathcal{ℳ},\mathrm{\Gamma })$, we give a fixed point characterization of Voiculescu amenability of $(\mathcal{ℳ},\mathrm{\Gamma })$ in terms of modules over ${\mathcal{ℳ}}_{\ast }$. As a consequence, we present some descriptions for amenability of locally compact groups in terms of certain associated Hopf von Neumann algebras. We finally apply this result to some modules of continuous functions on a multiplicative subsemigroup of ${\mathcal{ℳ}}_{\ast }$.

In this paper, we construct derived equivalences between subalgebras of some $\mathrm{\Phi }$-Auslander–Yoneda algebras from a class of triangles in idempotent complete triangulated categories. The derived equivalences are obtained by transferring subalgebras induced by triangles to endomorphism algebras induced by approximation sequences.

We study the injectivity of the map (of pointed sets) $U{m}_{r+1}(R)/E_{r+1}(R)\to S{O}_{2(r+1)}(R)/E{O}_{2(r+1)}(R)$, which we had constructed in [12]. We obtain a necessary and sufficient condition for the kernel to be non-trivial under some additional assumptions on $r$.

In this paper, we first introduce the new class of LN rings, motivated by 2-primal rings and NI rings. We then deal with elementary properties of LN rings, observing relations between this and other known classes of rings. Finally, we study topological conditions, such as connectedness and normality, of several prime spectra that can be naturally associated to a given ring.

We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra $A$ as Hopf automorphisms, namely Molnar’s smash coproduct Hopf algebra. We find connections between the exponent and Frobenius–Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius–Schur indicators of the original Hopf algebra $A$. We study the category of modules of the smash coproduct.

Let $C$ be a semidualizing $R$-module, where $R$ is a commutative ring. We first introduce the definition of $C$-cotorsion modules, and obtain the properties of $C$-cotorsion modules. As applications, we give some new characterizations for perfect rings. Second, we study the Foxby equivalences between the subclasses of the Auslander class and that of the Bass class with respect to $C$. Finally, we discuss $C$-cotorsion dimensions and investigate the transfer properties of strongly $C$-cotorsion dimensions under almost excellent extensions.

For any positive integer $k$, let ${{𝒢}}_k$ denote the set of finite groups $G$ such that all Cayley graphs $\mathrm{\text{Cay}}(G,S)$ are integral whenever $\left|S\right|\le k$. Estélyi and Kovács [On groups all of whose undirected Cayley graphs of bounded valency are integral, Electron. J. Combin. 21 (2014) #P4.45.] classified ${{𝒢}}_k$ for each $k\ge 4$. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class ${{𝒢}}_3$ is characterized. As an application, the classification of ${{𝒢}}_k$ is obtained again, where $k\ge 4$.

In 1980, White conjectured that the toric ideal associated to a matroid is generated by binomials corresponding to a symmetric exchange. In this paper, we prove that classes of matroids for which the toric ideal is generated by quadrics and that has quadratic Gröbner bases, are closed under series and parallel extensions, series and parallel connections, and 2-sums.

Let $R$ be an associative ring with center $Z(R).$ The objective of this paper is to discuss the commutativity of a semiprime ring $R$ which admits a derivation $d$ such that $d([x^m,y^n])\pm [x^m,y^n]\in Z(R)$ for all $x,y\in R$ or $d([x^m,y^n])\in Z(R)$ for all $x,y\in R$ or $d(x^m\circ y^n)\in Z(R)$ for all $x,y\in R,$ where $m$ and $n$ are fixed positive integers. Finally, we apply these purely ring theoretic results to obtain commutativity of Banach algebra via derivation.

The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, J. Algebra 323 (2010) 2041–2057] to the bounded below case, is obtained. Moreover, we interpret the relative derived functor ${\mathrm{\text{Ext}}}_{{𝒳}{𝒜}}(-,-)$ as the morphisms in such derived category and give two applications.

Linear Ordinary Differential Equations (ODEs) with constant coefficients are studied by looking in general at linear recurrence relations in a module with coefficients in an arbitrary $\mathbb{Z}$-algebra. The bridge relating the two theories is the notion of formal Laplace transform associated to a sequence of invertibles. From this more economical perspective, generalized Wronskians associated to solutions of linear ODEs will be revisited, mentioning their relationships with Schubert Calculus for Grassmannians.

We study the situation where a solvable group $G$ has a faithful irreducible character $\chi$ such that $\chi \overline{\chi }$ has exactly two distinct nonprincipal irreducible constituents. We prove that $G$ has derived length bounded above by 8, and provide an example of such a group having derived length 8. In particular, this improves upon a result of Adan-Bante.

This is a study of Ding projective modules relative to a semidualizing module and related topics. Firstly, we study $D_C$-projective dimensions and $D_C$-projective modules under change of rings. Secondly, we establish a new version of the Foxby equivalence with respect to $D_C$-projective modules and $D_C$-injective modules. Thirdly, we characterize Ding projective modules in ${{𝒜}}_C(R)$ and Ding injective modules in ${{ℬ}}_C(R)$. At last, as applications, some new characterizations of perfect rings and quasi-Frobenius rings are given.

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring $R$ for which idempotents lift modulo the Jacobson radical $J(R)$, but idempotents do not lift modulo $J({𝕄}_2(R))$. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if $I$ and $J$ are ideals of $R$ for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over $I+J$. On the positive side, if $I$ and $J$ are enabling ideals in $R$, then $I+J$ is also an enabling ideal. We show that if $I⊴R$ is (weakly) enabling in $R$, then $I[t]$ is not necessarily (weakly) enabling in $R[t]$ while $I⟦t⟧$ is (weakly) enabling in $R⟦t⟧$. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

Consider a weak crossed product order $C_f:={\sum }_{\sigma \in G}S{x}_{\sigma }$ in ${\mathrm{\Sigma }}_f:={\sum }_{\sigma \in G}K{x}_{\sigma }$, where $S$ is the integral closure of a discrete valuation ring $R$ in a tamely ramified Galois extension $K$ of the field of fractions of $R$. Assume that $S$ is local. In this paper, we show that $C_f$ is hereditary if and only if it is maximal among the weak crossed product orders in ${\mathrm{\Sigma }}_f$. We also give an algorithm that constructs, in terms of the basis elements $x_{\sigma }$ and the cocycle $f$, the unique hereditary weak crossed product order in ${\mathrm{\Sigma }}_f$ that contains a given $C_f$, and we give a criterion for determining whether that hereditary order will have a cocycle that takes nonunit values in $S$.

We study classical $R$-matrices $D$ for Lie algebras $𝔤$ such that $D$ is also a derivation of $𝔤$. This yields derivation double Lie algebras $(𝔤,D)$. The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

The b-function and related invariants of the singularity $z_1^4+z_2^p+z_1{z}_2^{p-1}=0$ are determined. We exploit the theory of computer algebra and Weyl algebra for the study of local b-function by T. Yano in 1970s.

In this paper we give a nice formula for the Castelnuovo–Mumford regularity of the Segre product of modules, under some suitable hypotheses. This extends recent results of David A. Cox, and Evgeny Materov (2009).

Let $R$ be a commutative Noetherian local ring. A finitely generated Gorenstein projective $R$-module $M$ is called IG-projective if there exists either an irreducible epimorphism $P\to M$ or irreducible monomorphism $M\to P$ with $P$ being a projective $R$-module. In this paper, we give a note on Gorenstein Projective Conjecture: a finitely generated IG-projective module is projective if and only if it is self-orthogonal.

Let $G$ be a finite group and $H$ be a subgroup of $G$. The non-normal graph of $H$ in $G$, denoted by ${𝔑}_{H,G}$, is defined as the bipartite graph with two parts $G\backslash N_G(H)$ and $H\backslash H_G$, where $N_G(H)$ and $H_G$ are the normalizer and the core of $H$ in $G$, respectively. Two vertices $v\in H\backslash H_G$ and $u\in G\backslash N_G(H)$ are adjacent if $u^{-1}vu\notin H$. In this paper, we consider vertex and edge connectivity of ${𝔑}_{H,G}$. We show that ${\kappa }^{\prime }({𝔑}_{H,G})=\delta$ and if $k$ is a positive integer such that $k\le \left|H\backslash H_G\right|$ and $\left|H\right|>2$, then the graph ${𝔑}_{H,G}$ has a cycle of length $2k$.