This paper is concerned with
Hochschild’s “maximal algebra” which has also been discussed by Jans, Jans and
Nakayama, and Zaks.
A category of rings for which the maximal algebra construction is valid is first
defined. These are the “split rings” of the title. These rings include the split rings
introduced by Jans and Nakayama but they need not be semi-primary. A second
category consisting of a kind of sheaf over a directed graph is introduced. Using this
second category the maximal algebra construction is exhibited as a composition of
adjoint functors, and hence gives the universal mapping property of the maximal
algebra. The properties of the sheaf category are then used to show that
a semi-primary split ring is the image of a semi-primary ring of a special
type.
