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Free Modules with Isomorphic Duals by Kyriopoulos, Theodoros is a scholarly article available to read on EtoBox.

What is Free Modules with Isomorphic Duals about?

Let M, N be free modules over a Noetherian commutative ring R and let F be a field whose cardinality does not exceed the continuum. We prove the following : 1) The assertion that [Any two F-vector spaces with isomorphic duals are isomorphic] is equivallent to the ICF (Injective continuum 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) If M, N have isomorphic duals then they are themselves isomorphic (over R), when rank(M) is not an {\omega}-measurable cardinal and R is a non-Artinian ring that is either Hilbert or countable. 5) If R is a non-local domain then R is a half-slender ring. 6) If R is Artinian and 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. We also prove that if the powersets of two given sets have equal cardinalities then there is a bijection from the one powerset to the other that preserves the symmetric difference of sets.

Author

Kyriopoulos, Theodoros

Published

2024

Language

EN