The Characteristic of the Minimal Ideals and the Minimal Generalized Ideals in Rings

In this paper, we prove that the characteristic of a minimal ideal and a minimal generalized ideal, which is meant to be one of minimal left ideal, minimal right ideal, bi-ideal, quasi-ideal, and m , n -ideal in a ring, is either zero or a prime number p . When the characteristic is zero, then the minimal ideal (minimal generalized ideal) as additive group is torsion-free, and when the characteristic is p , then every element of its additive group has order p . Furthermore, we give some properties for minimal ideals and for generalized ideals which depend on their characteristics.