Finite-dimensional square-free K-algebras have been completely characterized by Anderson and D'Ambrosia as certain semigroup algebras A ∼ = K ξ S over a square-free semigroup S twisted by some ξ ∈ Z 2 (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 2 (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 α ξ S there exists a short exact sequenceconnecting the outer automorphisms of R to certain cohomology groups related to S and D.

Abstract. We compute the monoid V[L K (E)] of isomorphism classes of finitely generated projective modules of a Leavitt path algebra over an arbitrary directed graph. Our result generalizes the result of Ara, Moreno, and Pardo in which they computed the monoid V[L K (E)] of a Leavitt path algebra over a countable row-finite directed graph.