A class of rings with the 2-sum property

Abstract

Recall that a ring satisfies the 2-sum property if each of its elements is a sum of two units. Here a ring R is said to satisfy the binary 2-sum property if, for any a, b in R, there exists a unit u of R such that both $$a-u$$ and $$b-u$$ are units. A well-known result, due to Goldsmith, Pabst and Scot, states that a semilocal ring satisfies the 2-sum property iff it has no image isomorphic to $$\mathbb {Z}_2$$ . It is proved here that a semilocal ring satisfies the binary 2-sum property iff it has no image isomorphic to $${\mathbb {Z}}_2$$ or $${\mathbb {Z}}_3$$ or $${\mathbb {M}}_2({\mathbb {Z}}_2)$$ .