Rank of elements of general rings in connection with unit-regularity
Nik Stopar
TL;DR
This paper introduces a new concept of element rank in general rings, explores its properties, and connects it to unit-regularity, providing characterizations and applications in semiprime rings.
Contribution
It defines the rank of elements in unital rings, relates it to invertibility in semiprime rings, and proves all socle elements are unit-regular.
Findings
Rank of elements characterized in semiprime rings
Every socle element in a unital semiprime ring is unit-regular
Examples supporting the rank definition in various rings
Abstract
We define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we prove that every element in the socle of a unital semiprime ring is unit-regular.
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