6 K. R. Goodearl and A. K. Boyle
nilpotent right ideal of T. According to 1.7* T is a regular
ring, hence we obtain eT(l-e) = 0. For any t e T, we thus have
(l-e)te = et(l-e) = 0, whence te = ete = et. Therefore e is
a central idempotent in T.
COROLLARY 1.10, Let R be a regular, right self-injective ring.
If I is any two-sided ideal of R, then the S-closure of I- p in
R~ is a two-sided ideal generated by a central idempotent.
Proof. Inasmuch as RR e 7((R), this result is immediate from 1.9»
PROPOSITION 1,11, 7J(R) is an abelian category.
Proof. We have noted that 7?(R) has finite products and
coproducts, and it is clear from 1#3 and 1,5 that 7?(R) has kernels
and cokernels. Any subobject of an object A e 7?(R) is a direct
summand of A and thus is the kernel of a projection p : A - » A.
Likewise, every quotient object in 7?(R) is a cokernel.
PROPOSITION 1,12. If {A.} is any set of objects in 7?(R),
then TTA. is a product of the A. in 7l(R), and E(®A.) is a
coproduct of the A. in 7?(R).
Proof. Since TTA- is a nonsingular injective module, it clearly
is a product of the A. in 7?(R).
Inasmuch as €A. is a nonsingular essential submodule of
E(€A.), we see that E(®A.) is nonsingular, whence E(©A.) e 7?(R).
For each j, let k, :A. -E(®A.) denote the natural injection.
Given B e 7l(R) and maps f. : A. - » B for all i, we obtain a map
f : ©A. -*B, which by injectivity extends to a map g : E(©A.) - B
such that gk. = f , far all j. If we have another map
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