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.