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 has an associated division ring D and square-free semigroup S. We show a square-free ring R can be characterized as a semigroup ring over a square-free semigroup S twisted by some (α, ξ) ∈ Z(S,D), a two-dimensional cocycle of S with coefficients in the nonabelian group of units D∗ of a division ring D. Also, to each square-free ring R ∼= D ξ S there exists a short exact sequence 1 −→ H (α,ξ)(S,D ∗) −→ Out R −→ Stab(α,ξ)(Aut S) −→ 1. connecting the outer automorphisms of R to certain cohomology groups related to S and D. Introduction The study of outer automorphisms of certain algebraic structures, beginning with Stanley [13] and continued by Scharlau [10], Baclawski [4], and Coelho [6], led to a characterization of the outer automorphism group of square-free algebras by Anderson and Dambrosia [2]. There, Clark’s [5] cohomology of semigroups, was adapted to prove to every square-free algebra there is an associated short exact sequence relating the first cohomology group with the outer automorphisms of the algebra. Later, their approach was adopted by Sklar [11] to obtain results about the class of binomial algebras, which includes the class of square-free algebras. Square-free rings were defined by D’Ambrosia [7] as a generalization of square-free algebras. She developed the basic properties of square-free rings and provided examples of square-free rings that are not of the form D ⊗K A, where D is a division ring with center K and A is a square-free K-algebra. Date: September 29, 2008. 2000 Mathematics Subject Classification. Primary: 16P20.