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 $\left(m,n\right)$ -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.