Walker's Cancellation Theorem |
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Authors: | Robert Lubarsky |
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Institution: | Department of Mathematics , Florida Atlantic University , Boca Raton , Florida , USA |
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Abstract: | Walker's cancellation theorem says that, if B ⊕ Z is isomorphic to C ⊕ Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category. |
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Keywords: | Abelian groups Cancellation Constructive mathematics Direct sums Diagram category |
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