Abstract. A ring R is called 2-good if each element in R is a sum of two units. It was proved independently by both Wolfson and Zelinsky that the ring of linear transformations of a vector space V over a division ring D is 2-good except when V is one-dimensional over ℤ₂. A ring R is called right self-injective if each right R-homomorphism from any right ideal of R to R can be extended to an endomorphism of R.

As the ring of linear transformations is a right self-injective ring the result of Wolfson and Zelinsky generated quite a bit of attention towards understanding which right self-injective rings are 2-good. Khurana and Srivastava completely characterized right self injective 2-good rings by proving that a right self injective ring is 2-good if and only if it has no factor isomorphic to ℤ₂.

A ring R is called twin-good if for each element a ∊ R there exists a unit u ∊ R such that both a+u and a-u are units. Clearly every twin good ring is 2-good but converse need not be true. We prove (jointly with Professor Srivastava), that if R is a right self-injective ring then R is twin good if and only if R has no factor isomorphic to ℤ₂ or ℤ₃.