Square-free rings

Square-free algebras have been studied in a variety of settings and have been completely characterized by Anderson and D'Ambrosia. In this paper, we extend the definition of square-free to Artinian rings with identity and begin to develop the theory of square-free rings. We show that every square-free ring has an associated division ring and square-free semigroup. Many square-free rings are square-free D-algebras, i.e. rings of the form D circle times (K) A where D is a division ring with center K and A is a square-free K-algebra. We present a characterization of all square-free D-algebras and provide examples of square-free rings that are not D-algebras.