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.
Previous Page Next Page