Rings with Zassenhaus Families of Ideals

Abstract

Let R be a ring with identity. We call a family ℱ of left ideals of R a Zassenhaus family if the only additive endomorphisms of R that leave all members of ℱ invariant are the left multiplications by elements of R. Moreover, if R is torsion-free and there is some left R-module M such that R ⊆ M ⊆ R⊗ℤℚ and End ℤ(M) = R we call R a “Zassenhaus ring”. It is well known that all Zassenhaus rings have Zassenhaus families. We will give examples to show that the converse does not hold even for torsion-free rings of finite rank.