Free modules with isomorphic duals

TL;DR

This paper investigates properties of free modules with isomorphic duals over various rings, linking module isomorphisms to set-theoretic hypotheses and ring characteristics.

Contribution

It establishes new equivalences and conditions for free modules with isomorphic duals, connecting module theory with set theory and ring properties.

Findings

Dual isomorphism relates to the ICF hypothesis and is undecidable in ZFC.

If the dual of a free module is projective with infinite rank, then the ring is Artinian.

For Artinian rings with small cardinality, the dual of a free module is free.

Abstract

Let M, N be free modules over a Noetherian commutative ring R and let F be a field such that card(F) does not exceed the continuum. Then : (1) The assertion that [Any two F-vector spaces with isomorphic duals are isomorphic] is equivallent to the ICF (Injective continium function) hypothesis and it is a non-decidable statement in ZFC. (2) If the dual of M is a projective R-module and rank(M) is infinite then the ring R is Artinian. (3) If R is Artinian and card(R) does not exceed the continuum then the the dual of M is free. (4) Assume that R is a non-Artinian ring that is either Hilbert or countable. Then : (a) If M, N have isomorphic duals then they are themselves isomorphic (b) Any free direct summand of the dual of M is finitely generated, if Card(R) is not omega-measurable. (c) If R is connected and both Card(R), rank(M) are not omega-measurable then [Any direct summand of the dual of M that is not finitely generated is a dual of a free R-module]. (5) If R is a non-local domain then R is a half-slender ring. (6) If R is Artinian ring and it's cardinality card(R) does not exceed the continuum then the assertion that [any two free R-modules with isomorphic duals are isomorphic] is non-decidable in ZFC. (7) If R is a domain and rank(M) is infinite then the Goldie dimension of the dual of M is equal to it's cardinality . (8) If R is a complex affine algebra whose corresponding affine variety has no isolated points then [any two projective R-modules with isomorphic duals are themselves isomorphic]. (9) Let V be an F-vector space of infinite dimension . The dimension of it's dual is cardinality of the powerset of the the dimension of V.