Unit sum numbers of right self-injective rings

Abstract

In a recent paper (which is to appear in J. Algebra Appl.) we proved that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to ℤ2 and hence the unit sum number of a nonzero right self-injective ring is 2, ω or ∞. In this paper we characterise right self-injective rings with unit sum numbers ω and ∞. We prove that the unit sum number of a right self-injective ring R is ω if and only if R has a factor ring isomorphic to ℤ2 but no factor ring isomorphic to ℤ2 × ℤ2, and also in this case every element of R is a sum of either two or three units. It follows that the unit sum number of a right self-injective ring R is ∞ precisely when R has a factor ring isomorphic to ℤ2 × ℤ2. We also answer a question of Henriksen (which appeared in J. Algebra, Question E, page 192), by giving a large class of regular right self-injective rings having the unit sum number ω in which not all non-invertible elements are the sum of two units.