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 [
Citation14], Baclawski [
Citation4], Clark [
Citation5], Coelho [
Citation6], Anderson and D'Ambrosia [
Citation1], have produced characterizations of square-free algebras. Here using the non-abelian cohomology of Dedecker [
Citation8] 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
![](//:0)
, 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 [
Citation6]) of automorphism groups to prove that if
![](//:0)
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.