AbstractThe object of this paper is to define a general regularity for Γ-rings and to explore ways of generating such regularities. Two main approaches are identified. Regularities for Γ-rings can be generated by means of polynomials. These methods are the main topic of Section 2 where the concepts of polynomial, ideal generated polynomial and ideal generated regularities are introduced. The second approach is via existing regularities for ring which were introduced by Roos. All known regularities for Γ-rings turn out to special cases of at least one type of regularity introduced in Section 2. The Behrens radical for Γ-rings is introduced in Section 4 and is characterized using an ideal-generated regularity for Γ-rings.