8 K. R. Goodearl and A. K. Boyle
shows that B is the supremum in L*(E(®A,)) of all Bf1E( @ A.),
i e J
Gabriel and Oberst have shown that all spectral categories have
the form ^(R), as follows,
THEOREM 1^4. A category ^ is a spectral category if and only
if there exists a regular, right self-injective ring R such that
7h is equivalent to 7|(R).
Proof. According to [3, Satze 2.1,2.2], ^ is a spectral
category if and only if there exists a regular, right self-injective
ring R such that V\ is equivalent to the full subcategory
9 1 £ Mod-R generated by all direct summands of powers of IL .
Inasmuch as R is regular and right self-injective, we see that
RR e ??(R), from which it follows that S I Q 7l(R). On the other hand,
[4, Proposition 1.16] says that R = S°R and then [4, Proposition
1.18] shows that any nonsingular right R-module can be embedded in
a power of RR , whence 7J(R) C 51. Therefore H = 7?(R).
In particular, it follows from 1.15 and 1.14 that for any ring
R there must exist a regular, right self-injective ring Q such
that 7?(R) is equivalent to 7?(Q). Actually, we can find such a Q
such that 7KR) is equal to ??(Q), i.e., so that ??(R) and 7?(Q)
have the same objects, the same morphisms, and the same law of
THEOREM 1.15* There exists a regular, right self-injective ring
Q such that 7l(R) = WQ).