A Class of Ring-like Objects

TL;DR

This paper introduces new algebraic structures called left dirings and their associated modules, ideals, and radicals, providing characterizations and properties that extend classical ring theory concepts.

Contribution

It defines novel concepts such as one-sided dirings and 3-primitive dirings, and characterizes 3-semi-primitive dirings and 3-radicals, expanding the algebraic framework.

Findings

Characterization of 3-semi-primitive left dirings

External characterizations of 3-semi-primitive dirings

Description of the 3-radical via 3-primitive ideals

Abstract

We introduce the notions of one-sided dirings, 3-irreducible left modules, 3-primitive left dirings, 3-semi-primitive left dirings, 3-primitive ideals and 3-radicals. The main results consists of two parts. The first part establishes two external characterizations of a 3-semi-primitive left diring. The second part characterizes the 3-radical of a left diring by using 3-primitive ideals.