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.

Key Words:

• Automorphism
• Nonabelian cohomology
• Square-free rings

2000 Mathematics Subject Classification:

• Primary 16P20

ACKNOWLEDGMENT

The author would like to thank Professor Frank Anderson, for his continued guidance and inspiration, and the referee, who gave many helpful comments and suggestions.

Notes

Communicated by D. Zacharia.