International Journal of Algebra and Computation

Given a partial action $\alpha$ of a connected groupoid ${𝒢}$ on an associative ring $A$ we investigate under what conditions the partial skew groupoid ring $A{\star }_{\alpha }{𝒢}$ can be realized as a partial skew group ring. In such a case, applications concerning to the separability, semisimplicity and Frobenius property of the ring extension $A\subset A{\star }_{\alpha }{𝒢}$ as well as to the artinianity of $A{\star }_{\alpha }{𝒢}$ are given.

By an additive action on an algebraic variety $X$ we mean a regular effective action ${𝔾}_a^n\times X\to X$ with an open orbit of the commutative unipotent group ${𝔾}_a^n$. In this paper, we give a classification of additive actions on complete toric surfaces.

We find a semigroup $Q_n$, whose category of partial representations contains the representation category $Rep({{ℱ}}_n)$ of the free left regular band ${{ℱ}}_n$. We use this to construct a resolution for the absolute kernel of a representation of ${{ℱ}}_n$, for which the kernel ${\mathrm{\text{Sp}}}_n$ of the Markov operation in the Tsetlin library model is a prominent example. We obtain a formula for the dimension of the absolute kernel, generalizing the equality of the dimension of ${\mathrm{\text{Sp}}}_n$ to the number of derangements of order $n$.

In this paper, we present new characterizations of several radicals of differential polynomial rings, including the Levitzki radical, strongly prime radical, and uniformly strongly prime radical in terms of the related $\delta$-radical.

In this paper, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous paper, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic and acylindrically hyperbolic.

Let $p$ be an odd prime and let $J_o(X)$, $J_r(X)$ and $J_e(X)$ denote the three different versions of Thompson subgroups for a $p$-group $X$. In this paper, we first prove an extension of Glauberman’s replacement theorem [G. Glauberman, A characteristic subgroup of a p-stable group, Canad. J. Math. 20 (1968) 1101–1135, Theorem 4.1]. Second, we prove the following: Let $G$ be a $p$-stable group and $P\in {\mathrm{\text{Syl}}}_p(G)$. Suppose that $C_G(O_p(G))\le O_p(G)$. If $D$ is a strongly closed subgroup in $P$, then $Z(J_o(D))$, $\mathrm{\Omega }(Z(J_r(D)))$ and $\mathrm{\Omega }(Z(J_e(D)))$ are normal subgroups of $G$. Third, we show the following: Let $G$ be a $\mathrm{\text{Qd}}(p)$-free group and $P\in {\mathrm{\text{Syl}}}_p(G)$. If $D$ is a strongly closed subgroup in $P$, then the normalizers of the subgroups $Z(J_o(D))$, $\mathrm{\Omega }(Z(J_r(D)))$ and $\mathrm{\Omega }(Z(J_e(D)))$ control strong $G$-fusion in $P$. We also prove a similar result for a $p$-stable and $p$-constrained group. Finally, we give a $p$-nilpotency criteria, which is an extension of Glauberman–Thompson $p$-nilpotency theorem.

For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where ${〈x〉}^G$ is not nilpotent.

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic $n\times n$ matrices. We establish that in case $0<p\le n$ the maximal degree of indecomposable invariants tends to infinity as $d$ tends to infinity. In other words, there does not exist a constant $C(n)$ such that it only depends on $n$ and the considered algebra of invariants is generated by elements of degree less than $C(n)$ for any $d$. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of $p$ the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese type, squarefree Veronese ideals or $t$-spread Veronese ideals. In this paper we focus on $c$-bounded $t$-spread Veronese ideals and on Veronese ideals of bounded support. Their powers as well as their fiber cone are also considered.

We discuss two corrections concerning the paper mentioned in the title. One of them pertains to a slight error in a statement from a different paper that was used. For this statement, it is explained how to fix it and why this does not affect the correctness of our results. The other is concerned with an error in the original GAP source code used to verify some of our results. This error, which was subsequently fixed, did affect the correctness of one of our main theorems, which is now given in corrected (and stronger) form.