When Is Every Non Central-Unit A Sum of Two Nilpotents?

Abstract

A ring is said to satisfy the 2 2 -nil-sum property if every non central-unit is the sum of two nilpotents. We prove that a ring satisfies the 2 2 -nil-sum property iff it is either a simple ring with the 2 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 2 -nil-sum property that is not commutative. Moreover, a simple right Goldie ring has the 2 2 -nil-sum property iff it is a field.