Right Self-injective Rings in Which Every Element Is a Sum of Two Units

In 1954 Zelinsky [16] proved that every element in the ring of linear transformations of a vector space V over a division ring D is a sum of two units unless dim V = 1 and D = Z2. Because EndD(V ) is a (von-Neumann) regular ring, Zelinsky’s result generated quite a bit of interest in regular rings that have the property that every element is a sum of (two) units. Clearly, a ring R, having Z2 as a factor ring, cannot have every element as a sum of two units. In 1958 Skornyakov [12, Problem 31, p. 167] asked: Is every element of a regular ring (which does not have Z2 as a factor ring) a sum of units? This question of Skornyakov was settled by Bergman (see [7]) in negative who gave an example of a directly-finite, regular ring with 2 invertible, in which not all elements are the sum of units. It is easy to see that if R is a unit regular ring with 2 invertible, then every element can be written as a sum of two units (see [3]). A number of authors have also considered arbitrary rings in which elements are the sum of units. For instance, Henriksen in [8, Theorem 3] proved that, an arbitrary ring R, every element of Mn(R), n > 1, is a sum of three units. Henriksen also gave an example of a ring R such that not every element of M2(R) is a sum of two units [8, Example 10].