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

Abstract

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.