The concepts of closed submodule, left and right annihilators are generalised and a necessary and sufficient condition on any Morita context is given so that the duality and projectivity between these sets exist. As a special case, we get Hutchinson's necessary and sufficient condition under a weaker hypothesis on the context. Let U be a left A-module and S the endomorphism ring of U. In 1977 [6], under the assumption that U is projective and contains a unimodular element, a proof was given

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... there is an order inverting bijection called duality, and an order preserving bijection called projectivity between the closed submodules of U and the right, respectively, left annihilators in S of subsets of 5 . In 1987 [3], in the more general setting of a Morita context (R, U, V, S), under a much weaker hypothesis that Us is faithful and (U,V) = R, which is equivalent to (R,U,V,S) being nondegenerate [1, Theorem 2] and (U, V) = R, Hutchinson proved that the duality and projectivity exist if and only if U is torsionless. Since a projective module is torsionless, these results generalise the theorems in [6], and since necessary and sufficient conditions are given, these results are in some sense the best. In this paper, we generalise the concepts of closed submodule, and left and right annihilators, and give a necessary and sufficient condition on any Morita context so that the duality and projectivity between these sets exist. As a special case, we get Hutchinson's necessary and sufficient condition under a weaker hypothesis on the context. Throughout this paper, R and S are associative rings with identity, and R-Mod, S-Mod denote respectively the categories of unital left R-, and left 5-modules. Modules, unless otherwise specified, are consistently left modules. Recall for a Gabriel topology T, a module M is called T-torsion if Ann ji(m) £ T for every m € M, and any module M has a largest r-torsion submodule T r (Af); a module M is called T-free if Ann^f {A) = 0 for every A € T; a submodule K of M is called r-saturated if M/K is T-free, and T T (M) is the smallest r-saturated submodule of M. For any submodule K of M, K = {m G M, Am C K for some A E T} is