Square-Free Rings and Their Automorphism Groups

Abstract

A square-free ring is an artinian ring in which each indecomposable projective module has no repeated composition factors. Such square-free rings are closed under Morita equivalence. All square-free algebras, those finite dimensional algebras A over a field K with the property that for every pair of primitive idempotents of A, are square-free as rings and include all incidence algebras of posets over fields. Several earlier studies, including ones by Stanley [14], Baclawski [4], Clark [5], Coelho [6], Anderson and D'Ambrosia [1], have produced characterizations of square-free algebras. Here using the non-abelian cohomology of Dedecker [8] we generalize a characterization [21] of square-free algebras by showing that an indecomposable, basic artinian ring R is square-free iff it is isomorphic to a ring , that is constructed as the vector space DS over a division ring D with basis a square-free semigroup S where multiplication is twisted by a 2-cocycle (α, ξ) of S with coefficients in the division ring D. We then generalize studies (see [6]) of automorphism groups to prove that if is a square-free ring, then there is a short exact sequence where W is the stabilizer of the action of (α, ξ) on Aut(S), and when (α, ξ) is trivial, W = Aut(S) and the sequence splits.