Journal of Algebra and Its Applications

The purpose of this paper is to explore new aspects of the prime ideal structure of tensor products of algebras over a field k. We prove that given a k-algebra A and a normal field extension K of k (in the sense of Galois theory), then for any prime ideals P₁ and P₂ of K ⊗_k A lying over a fixed prime ideal p of A, there exists a k-automorphism σ of K such that (σ ⊗_kid_A)(P₁) = P₂. As an Application, we establish a result related to the dimension theory of tensor products stating that, for two arbitrary k-algebras A and B, the minimal prime ideals of p ⊗_k B + Aσ_k q have the same height, for any prime ideals p and q of A and B, respectively.

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

Our main result here is the following theorem: Let G = AT, where A is a Hall π-subgroup of G and T is p-nilpotent for some prime p ∉ π, let P denote a Sylow p-subgroup of T and assume that A permutes with every Sylow subgroup of T. Suppose that there is a number p^k such that 1 < p^k < |P| and A permutes with every subgroup of P of order p^k and with every cyclic subgroup of P of order 4 (if p^k = 2 and P is non-abelian). Then G is p-supersoluble.

A group G is said to be a B(n,k) group if for any n-element subset {a₁,…, a_n} of G, |{a_ia_j | 1≤i, j≤n}|≤ k. In this paper, characterizations of B(4,11) groups, B(4,12) groups, and B(4,13) non-2-groups are given.

A (commutative integral) domain R is said to be valuative if, for each nonzero element u in the quotient field of R, at least one of R ⊆ R[u] and R ⊆ R[u^-1] has no proper intermediate rings. Such domains are closely related to valuation domains. If R is a valuative domain, then R has at most three maximal ideals, and at most two if R is not integrally closed. Also, if R is valuative, the set of nonmaximal prime ideals of R is linearly ordered, at most one maximal ideal of R does not contain each nonmaximal prime of R, and R_P is a valuation domain for each prime P except for at most one maximal ideal. Any integrally closed valuative domain is a Bézout domain. Valuation domains are characterized as the quasilocal integrally closed valuative domains. Each one-dimensional Prüfer domain with at most three maximal ideals is valuative.

We consider a class of self-injective special biserial algebras Λ_N over a field K and show that the Hochschild cohomology ring of dΛ_N is a finitely generated K-algebra. Moreover, the Hochschild cohomology ring of Λ_N modulo nilpotence is a finitely generated commutative K-algebra of Krull dimension two. As a consequence the conjecture of [N. Snashall and Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc.88 (2004) 705–732], concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras.

The main results of this paper are two infinite families of series-product identities which are based on a classical Gauss identity and two different interpretations of characters of fundamental modules for the affine Kac–Moody Lie algebra , one of them stemming from a boson–fermionic realization of the affine Lie algebra .

We introduce and study comodules over cowreath defined over a given coring. If our coring arises from an entwining structure, we then give a procedure to construct a cowreath from a given cowreath over the factor coalgebra. We also include the dual notions, that is, wreaths over ring extension and their modules. In particular, we show that the study of twisted algebras and twisted bimodules already introduced in [A. Čap, H. Schichl and J. Vanžura, On twisted tensor products of algebras, Commun. Algebra23 (1995) 4701–4735] has its origin in the study of wreath and their bimodules.