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A ring is said to satisfy the $2$-nil-sum property if every non central-unitis the sum of two nilpotents. We prove that a ring satisfies the $2$-nil-sumproperty iff it is either a simple ring with the $2$-nil-sum property or acommutative local ring with nil Jacobson radical, and we provide an example ofa 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.

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