When is every non central-unit a sum of two nilpotents?

TL;DR

This paper characterizes rings where every non-central unit can be expressed as a sum of two nilpotents, showing such rings are either simple with this property or commutative local with nil Jacobson radical.

Contribution

It provides a complete characterization of rings satisfying the 2-nil-sum property, including examples and conditions for simple right Goldie rings.

Findings

A ring satisfies the 2-nil-sum property iff it is simple with the property or a commutative local ring with nil Jacobson radical.

A simple ring with the 2-nil-sum property can be non-commutative, with an explicit example provided.

A simple right Goldie ring has the 2-nil-sum property iff it is a field.

Abstract

A ring is said to satisfy the $2$-nil-sum property if every non central-unit is the sum of two nilpotents. We prove that a ring satisfies the $2$-nil-sum property iff it is either a simple ring with the $2$-nil-sum property or a commutative local ring with nil Jacobson radical, and we provide an example of a simple ring with the $2$-nil-sum property that is not commutative. Moreover, a simple right Goldie ring has the $2$-nil-sum property iff it is a field.