NONSINGULAR INJECTIVE MODULE
PROPOSITION 1.8. Let C e 7((R), and set T = End^C). There
is a lattice isomorphism of L*(T ) onto L*(C), given by the
rule J B»JC.
Proof. We observe from 1.7 that TT e W D .
In view of 1.3» a^y J e L*(T ) must have the form eT for
some idempotent e, whence JC = eC e L*(C). Thus the rule J ^ JC
defines a monotone map ^ L*(Trp) "*L*(C). Given any A e L*(C),
1.3 shows that A = eC for some idempotent e e T, whence
eT e L*(T ) and 4(eT) = A. Therefore j is surjective.
If ^ is an isomorphism of partially ordered sets, it must also
be a lattice isomorphism. Inasmuch as ^ is surjective, all that
remains is to prove that for all J,K e L*(T ), J ^ K if and only
if i(J) ^ jU).
Thus suppose that J,K e L*(TT with ((J)^ j(K). Then there
exist idempotents e,f e T such that eT = J, fT = K, and
eC ^ fC. As a result, e = fe, whence J ^ K. Therefore J % K
if and only if 4(J) ^ 4(K),
as
required.
PROPOSITION 1.9« If C e 7|(S) and A is a fully invariant
submodule of C, then there exists a central idempotent e e End^(C)
such that eC is the ^-closure of A in C.
Proof. The S-closure of A in C must have the form eC for
some idempotent e in the ring T = End^(C). For any t e T, there
is a monomorphism of C/t~ (eC) into the nonsingular module C/eC,
whence t" (eC) e L*(C). Inasmuch as tA ^ A ^ eC, we have
A ^ t""1(eC), from which it follows that eC ^ t~X(eC) and
consequently teC ^ eC. Thus te = ete.
Now (1-e)Te = 0, from which we infer that eT(l-e) is a
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