Group rings of free products are primitive

Primitive central idempotents of finite group rings of symmetric groups

Let p be a prime. We denote by Sn the symmetric group of degree n, by An the alternating group of degree n and by Fp the field with p elements. An important concept of modular representation theory of a finite group G is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring FqG, where q is a prime...

Generalized Primitive Elements of a Free Group

We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts “like an automorphism” on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological prope...

Quasi-Primitive Rings and Density Theory

Square-free Rings and Their Automorphism Group

Finite-dimensional square-freeK-algebras have been completely characterized by Anderson and D’Ambrosia as certain semigroup algebras A ∼= KξS over a square-free semigroup S twisted by some ξ ∈ Z (S,K), a two-dimensional cocycle of S with coefficients in the group of units K∗ of K. D’Ambrosia extended the definition of square-free to artinian rings with unity and showed every square-free ring ha...

Cohomology Rings of Almost-direct Products of Free Groups

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space. 1. Almost direct products of free...