The Existence of Dual Modules

In this note we show that a Noetherian module has a dual module if and only if it satisfies AB5*. A connection between completeness and AB5* is also established. In this note we relate completeness, quasi-completeness, the A B5* condition, and duality. The main result is that a Noetherian R-module has a dual module if and only if it satisfies A B5*. Throughout this note R will denote a commutative ring with identity and all modules will be unitary. The terms local and semilocal will carry the Noetherian hypothesis. We use L(A) or LR(A) to denote the lattice of Rsubmodules of A. An R-module B is said to be a dual of an R-module A if there exists an order reversing lattice isomorphism 0: L(A) —> L(B) satisfying 0(JN) = 0(N): J for all R-submodules A of A and all ideals J of R. Any R-module A satisfies the so-called AB5 condition: for any submodule B and any ascending chain {Ba} of submodules of A, B n (L)aBa) = L)a(B fl Ba). A satisfies the dual condition AB5* if for any submodule B and any descending chain {Ba} of submodules, it follows that B + ((~\aBa_) = Da(B + Ba). Not every module satisfies AB5*; for example, Z, the integers, does not. However, any module having a dual necessarily satisfies AB5*. We show that for Noetherian modules, the converse is also true. We first show that the condition AB5* is closely related to completeness. Let R be a semilocal ring with Jacobson radical J and let A be a finitely generated .R-module. If A is complete in the 7-adic topology, it is well known [6] that A satisfies the condition (*) For any descending chain {Bn}%Lx of submodules of A and any integer k, there exists an integer n(k) such that Bn,k^ G (D„xLxBn) + JkA. A finitely generated module over a semilocal ring will be called quasicomplete if it satisfies (*). The first theorem relates the concepts ©f quasi-completeness and AB5*. Theorem 1. Let R be a semilocal ring and A a finitely generated R-module. Then A satisfies AB5* if and only if it is quasi-complete. Proof. Suppose A satisfies AB5*. Let {Bn}„*Lx be a countable descending chain of submodules of A and let k be a fixed integer. Then 00 CO JkA + Q Bn = Q (JkA + B„) = JkA + Bn(k) Received by the editors June 5, 1975. AMS (MOS) subject classifications (1970). Primary 13C05, 13E05, 13J10.