On small dual rings

TL;DR

This paper investigates small dual rings, characterizing their properties and relationships with dual rings, and applies these findings to provide new characterizations of QF rings.

Contribution

It introduces the concept of small dual rings, establishes their equivalence with dual rings under certain conditions, and generalizes known results in ring theory.

Findings

A ring is dual iff it is semilocal and small dual.

Properties of small dual rings are systematically explored.

New characterizations of QF rings are derived using small duality.

Abstract

A ring $R$ is called right (small) dual if every (small) right ideal of $R$ is a right annihilator. Left (small) dual rings can be defined similarly. And a ring $R$ is called (small) dual if $R$ is left and right (small) dual. It is proved that $R$ is a dual ring if and only if $R$ is a semilocal and small dual ring. Several known results are generalized and properties of small dual rings are explored. As applications, some characterizations of QF rings are obtained through small dualities of rings.