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