INTRODUCTION

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 [27]* 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

Kaplanskyfs

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

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