DECOMPOSING ELEMENTS OF A RIGHT SELF-INJECTIVE RING

Abstract

It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math.75 (1953) 358–386] and Zelinsky [Every linear transformation is sum of nonsingular ones, Proc. Amer. Math. Soc.5 (1954) 627–630] that every linear transformation of a vector space V over a division ring D is the sum of two invertible linear transformations except when V is one-dimensional over ℤ2. This was extended by Khurana and Srivastava [Right self-injective rings in which each element is sum of two units, J. Algebra Appl.6(2) (2007) 281–286] who proved that every element of a right self-injective ring R is the sum of two units if and only if R has no factor ring isomorphic to ℤ2. In this paper we prove that if R is a right self-injective ring, then for each element a ∈ R there exists a unit u ∈ R such that both a + u and a - u are units if and only if R has no factor ring isomorphic to ℤ2 or ℤ3.