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

as required.

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

composition.

THEOREM 1.15* There exists a regular, right self-injective ring

Q such that 7l(R) = WQ).