Lattice anti-isomorphisms of modules and the AB5* condition

Abstract

All rings considered in this paper are associative with an identity element and all modules are unitary. Homomorphisms of right modules will be wirtten on the left hand side. We shall use the following notation: L* is the dual of a partially ordered set L; End(X) and L(X) are the endomorphism ring and the submodule lattice of a module X.We recall some definitions and results. An element c of a complete lattice L isthere exists do e D such that c = do-A complete lattice L is called (weakly) algebraic if every element in L is a join of (weakly) compact elments. A complete lattice L is called upper continuous if a /\ (V</eD d) = Vj € ti( a A d) for all elements a e L and for all directed subsets D c L.The proofs of the following properties of an arbitrary complete lattice L may be found in [1]: 1) every compact element in L is weakly compact; 2) if L is upper continuous, then every weakly compact element in L is compact; 3) if L is algebraic, then L is upper contiunous. We have from these propositions Lemma 1. A complete lattice L is weakly algebraic and upper continuous if and only if L is algebraic.We now examine the lattices L (X) and L (X)*, where X is a module. The complete lattice L(X) is algebraic. Thereto an element Y of L{X) is compact if and only if Y is a finitely generated submodule of X. A submodule Y c X is called cofinite if Y is a weakly compact element of the complete lattice L(X)*. A module X is called finitely cogenerated if 0 is a cofinite submodule of X. It is easy to see that a submodule Y c X is cofinite if and only if the factor module X/Y is finitely cogenerated. We say a module X satisfies the property AB5* if the complete lattice L(X)* is upper continuous. As is well-known ([2], Lemma 1.4), the complete lattice L(X)* is weakly algebraic. Therefore a module X satisfies the AB5* condition if and only if L(X)* is algebraic.

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