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
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