Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I will state it again. Given two rings R_1 and R_2 (with or without identity. It's not specified.). If R_1[x] is isomorphic to R_2[y] (No such requirement that the isomorphism sends the constant terms to constant terms), can we deduce that R_1 iso. to R_2?

I feel there might be a counterexample but it's quite hard to find one.

share|improve this question
2  
This is the cancellation problem of Zariski. There are many counterexamples. See math.lsa.umich.edu/~hochster/Lip.text.pdf –  J.C. Ottem Dec 8 '10 at 14:19
    
This (interesting) question has received many good answers on math.SE: math.stackexchange.com/questions/13504/…. I have voted to close it here as "no longer relevant". –  Pete L. Clark Dec 9 '10 at 1:20
add comment

2 Answers

up vote 6 down vote accepted

This was recently asked and answered at http://math.stackexchange.com/questions/13504/does-rx-cong-sx-imply-r-cong-s As answered there, it is possible to find two non-isomorphic commutative rings whose polynomial rings in one variable are isomorphic. An example is given in http://www.ams.org/journals/proc/1972-034-01/S0002-9939-1972-0294325-3/home.html

share|improve this answer
add comment

Let X be an affine variety with two non-isomorphic vector bundles V and W that become isomorphic after adding a trivial line bundle to each. Then the coordinate rings of the total spaces of V and W should yield a counterexample. (Though you might have to do some extra work to verify that the total spaces of V and W are non-isomorphic as varieties.)

The counterexample cited by Tobias takes X to be the 2-sphere, V the tangent bundle to X, and W the trivial plane bundle over X.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.