A classical result of Zelinsky states that every linear transformation on a vector space V, except when V is one-dimensional over $Z_2$, is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if and only if no factor ring of R is isomorphic to $Z_2$. Some other related old and new results will also be discussed.