The basic purpose of this paper is to develop tools with which to study
the structure of an arbitrary regular, right self-injective ring R. [The
ubiquity and importance of such rings lie in the fact that the maximal right
quotient ring of any right nonsingular ring is regular and right self-
injective.] More specifically, we are concerned with the theory of types for
R, following Kaplansky [l^f] , and with developing dimension functions for the
lattice of principal right ideals of R, analogous to those of Loomis [l5] ^nd
von Neumann * Now the principal right ideals of a regular, right self-
injective ring R are also nonsingular infective right R-modules, and the
results we develop apply in fact to nonsingular infective modules over any ring.
Since no extra work is involved in developing the nonsingular infective theory
(actually, the proofs are simpler in some cases), we proceed in this generality.
Thus we develop the theory of types for nonsingular infective modules over any
ring, and we construct dimension functions which determine the isomorphism
classes of the nonsingular infective modules (in the sense that nonsingular
infective modules A and B are isomorphic if and only if each of the
dimension functions has the same value on A as on B).
We essentially borrow Kaplansky's definitions verbatim from [l*+] for the
theory of types, in the following manner. Given any nonsingular infective
module A, its endomorphism ring T is regular and self-injective. It follows
that T is a Baer ring in the sense of [l^-] , hence we quote
definitions for the possible types of T (finite, infinite, I, II, III, etc.).
[With two exceptions: to avoid ambiguity, we use "directly finite" and
"directly infinite" instead of "finite" and "infinite".] We then define the
type of A to be the same as the type of T, and translate these definitions
into more readily usable module-theoretic terms. While a sizable portion of
this part of our work could be derived as a consequence of Kaplansky*s theory,
we have taken a direct approach which works out in a more straightforward