Journal of Algebra and Its Applications

Given a group G acting on a ring R via α:G → Aut(R), one can construct the skew group ring R*_αG. Skew group rings have been studied in depth, but necessary and sufficient conditions for the simplicity of a general skew group ring are not known. In this paper, such conditions are given for certain types of skew group rings, with an emphasis on Von Neumann regular skew group rings. Next the results of the first section are used to construct a class of simple skew group rings. In particular, we obtain a more efficient proof of the simplicity of a certain ring constructed by J. Trlifaj.

Let (H, A) be a primitive central extension of a finite group G. We show that A is not characteristic in H in general, and further demonstrate in a series of examples that results, which hold about inner automorphisms of H, do not extend to the full automorphism group of H. We also give some new results about isoclinism and representation groups of G in the case that H is capable. Finally we give an example of two non-isomorphic groups of the same order, which not only have a representation group in common, but also have identical projective character tables.

Recent developments of Baxter algebras have led to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first and the second kinds, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras.

Let R be a ring and G a group of automorphisms of R. In this paper we investigate the transfer of the Von Neumman regularity from the total quotient ring of R to the total quotient ring of its fixed ring. We prove also that, for a locally finite group, Min(R^G) inherits the compactness from Min(R), where Min(R) denotes the set of all minimal prime ideals of R. Finally, We explore some conditions under which we can transfer the PF-property (respectively, pseudo PF-property) from a ring to its fixed subring.

We find a new curve of genus four attaining the Serre bound over prime fields. It is defined by the equation y¹²=x⁴(1-x), which attains the bound over the finite field if and only if the prime number p satisfies p ≡ 1 mod 12, and with an integer n. Furthermore, we show that if a standard conjecture of prime numbers is true then infinitely many prime numbers satisfy these conditions.

We introduce the notion of almost finite dimensional representability of algebras and study its connection with the classical finiteness conditions.

Let G be a finite group and H a subgroup of G. The Hecke algebra ℋ(G,H) associated with G and H is defined by the endomorphism algebra End_ℂ[G]((ℂ_H)^G), where ℂ_H is the trivial ℂ[H]-module and (ℂ_H)^G = ℂ_H⊗_ℂ[H] ℂ[G]. As is well known, ℋ(G,H) is a semisimple ℂ-algebra and it is commutative if and only if (ℂ_H)^G is multiplicity-free. In [6], by a ring theoretic method, it is shown that if the canonical involution of ℋ(G,H) is the identity then ℋ(G,H) is commutative and, if there exists an abelian subgroup A of G such that G = AH then ℋ(G,H) is commutative. In this paper, by a character theoretic method, we consider the commutativity of ℋ(G,H).

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

This paper continues the investigation of the theory of rational solutions of the CYBE for o(n) from the point of view of orders in the corresponding loop algebra, as it was developed in [8]. As suggested by [8], in the case of "singular vertices", we use the list of connected irreducible subgroups of SO(n) locally transitive on the Grassmann manifolds of isotropic k-dimensional subspaces in ℂⁿ obtained in [11]. New arguments based on the analysis of the structure of the stationary subalgebra of a generic point allow us to construct several rational solutions in o(7), o(8) and o(12).