manner than using [l^J would do, because our injectivity hypotheses are much

stronger than the Baer ring hypotheses in [l;-+]. The theory of types has also

been sketched out in the context of spectral categories by Roos [l8,20,22J,

and the theory of types for regular self-injective rings can be derived as a

consequence of his work. We have refrained from doing so, partly to minimize

categorical complications, and partly in order to obtain results applicable to

individual nonsingular infective modules.

The dimension theory falls into two distinct parts, which reflect the

basic differences between the "directly finite" and "directly infinite" cases.

[A nonsingular infective module is directly finite if it is not isomorphic to

any proper submodule of itself; otherwise it is directly infinite.] The

infinite dimension theory has no counterpart in Loomis or von Neumann. The

basic idea was developed in a special case by the first author in [5],hut to

our knowledge has not been developed elsewhere.

For the directly finite case, we develop a relative dimension theory

which is very similar in spirit to the dimension functions of Loomis [l5] and

von Neumann [27]• Actually, outside of a few exceptional cases, von Neumann's

dimension theory can be considered as a special case of ours, in a rather

roundabout fashion. Namely, von Neumann showed in [27] that nearly every

continuous geometry L could be realized as the lattice of principal right

ideals of a regular ring R, and Utumi showed in [26] that the continuity

conditions on L imply that R is self-injective. Thus our dimension theory,

restricted to the principal right ideals of R, yields the same dimension

function which von Neumann constructed directly on L. There is a major

difference between Loomis1 theory and ours: his theory applies to a lattice

with an orthocomplementation, whereas there is nothing in the general theory of

nonsingular infective modules to play the role of orthogonal complements.

However, our injectivity hypotheses are strong enough to overcome the lack of

orthogonal complements, and also to make our proofs somewhat easier than those

vi