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How to design or create or generate a bijective ring map?

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    $\begingroup$ I don't get the question. $\endgroup$
    – Martin Brandenburg
    Commented Sep 5, 2010 at 9:49
  • $\begingroup$ What's the actual question? $\endgroup$
    – Robin Chapman
    Commented Sep 5, 2010 at 10:36
  • $\begingroup$ We can validate ring map whether bijective or not, but inversely thinking where does these ring map come from? How to generate them is the actual question $\endgroup$
    – Simple carl
    Commented Sep 5, 2010 at 11:12

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In generality (this is tagged "commutative algebra", so let's talk commutative rings) I wonder if there is more than taking generators of each side and writing the images as polynomials in the generators of the other side. This is a candidate for a bijection of rings, but so far isn't a homomorphism (you need to check the relations hold). In other words express each ring as a quotient of a polynomial ring, set up homomorphisms from the polynomial rings, and then show the kernels are what they should be.

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  • $\begingroup$ It's great. Could you show an example in code of singular or macaulay. $\endgroup$
    – Simple carl
    Commented Sep 5, 2010 at 10:18

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