k K. R. Goodearl and A. K. Boyle
f is a monomorphism (epimorphism) in 7l(R) if and only if it is a
monomorphism (epimorphism) in Mod-R.
PROPOSITION 1,6. For any C e 7?(R), L*(C) is a complete,
complemented, modular lattice, with arbitrary infima given by
intersections. Finite suprema in L*(C) are given by sums, while
arbitrary suprema are given by S-closures of sums.
Proof. Inasmuch as L*(C) is closed under arbitrary intersections,
it is a complete lattice with all infima given by intersections. The
supremum of any collection {A.} c L*(C) is thus the intersection of
all ^-closed submodules of C which contain U A. , i.e., the
8-closure of £ A, in C. Given A , ...,A g L*(C), we have
A e...®A e 7&R) and a map A ®,,,®A - » C whose image is
A +... + A , hence 1.5 says that A_ + . ..+A cL*(C). Thus
1 n ^ 1 n
A +...+A is the supremum of {A , . ..,A } in L*(C).
Since the finite operations in L*(C) are given by sums and
intersections, we see that L*(C) is a modular lattice.
Given any A e L*(C), we have A@B = C for some B, and it is
clear that B is a complement for A in L*(C). Therefore L*(C)
is a complemented lattice.
The following theorem is usually attributed to Johnson and Utumi,
since it is proved in the same manner as [ 12, Theorem 2] and
[24, (4.6)]. A proof may be found in [4, Proposition 1.17].
THEOREM 1.7* I?or any A e 7?(R)» EndR(A) is a regular, right
self-injective ring.
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