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Math Help - Computing an isomorphism

  1. #1
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    Computing an isomorphism

    I am a programmer with some algebra knowledge so I can understand mathematical stuff but I can not really speak in maths, so please be indulgent. My issue is:
    Let R1(A,+,*) and R2(B,+,*) be two rings.
    1st question : is it ok to assume that at least an isomorphism exists between them considering the fact that the intersection of A and B is never void, or do they need to be identical in order for that assumption to be undoubtable.
    2nd question: at least intuitively do you think that an automated process given the right mathematical directives can find one isomorphism between those rings.
    3rd where can I find some clear info on matrix rings?

    Thanks
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: Computing an isomorphism

    Quote Originally Posted by jazzmoke View Post
    I am a programmer with some algebra knowledge so I can understand mathematical stuff but I can not really speak in maths, so please be indulgent. My issue is:
    Let R1(A,+,*) and R2(B,+,*) be two rings.
    1st question : is it ok to assume that at least an isomorphism exists between them considering the fact that the intersection of A and B is never void, or do they need to be identical in order for that assumption to be undoubtable.
    2nd question: at least intuitively do you think that an automated process given the right mathematical directives can find one isomorphism between those rings.
    3rd where can I find some clear info on matrix rings?

    Thanks
    The answer to your second question is "no". This is because the isomorphism problem for rings is, in general, insoluble. (Although a brute-force method would work for finite rings).

    I am unsure what your first question is asking. I mean, an isomorphism is entirely disjoint from the sets A and B which your rings are defined over. For example, one can take the ring generated by 3 in \mathbb{Z}/36\mathbb{Z}, and the ring generated by 2. These have different orders, and so are non-isomorphic. On the other hand, one can take the ring \mathbb{Z} and it is isomorphic to the ring consisting of 2-by-2 matrices,

    \left( \begin{array}{cc}x & 0 \\0 & x \end{array} \right)

    where x\in\mathbb{Z}. The sets you are working in don't matter!

    I think I would recommend you looking up the computer algebra package GAP. It often tells you how it does stuff in the manual.
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  3. #3
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    Re: Computing an isomorphism

    [QUOTE=jazzmoke;675274]I am a programmer with some algebra knowledge so I can understand mathematical stuff but I can not really speak in maths, so please be indulgent. My issue is:
    Let R1(A,+,*) and R2(B,+,*) be two rings.
    1st question : is it ok to assume that at least an isomorphism exists between them considering the fact that the intersection of A and B is never void, or do they need to be identical in order for that assumption to be undoubtable.[quote]
    Let R1 be the ring of integers and R2 the ring of rational numbers, both with the standard operations. Obviously, "the intersection of A and B is never void" because it is simply the integers but they are clearly not isomorphic. (Since R1 and R2 are fixed and do not change, "never" seems a strange word here.)

    2nd question: at least intuitively do you think that an automated process given the right mathematical directives can find one isomorphism between those rings.
    If two rings are isomorphic, then "given the right mathematical directives". Of course, that depends on knowing what "the right mathematical directives" are!

    3rd where can I find some clear info on matrix rings?

    Thanks
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Re: Computing an isomorphism

    If two rings are isomorphic, then "given the right mathematical directives". Of course, that depends on knowing what "the right mathematical directives" are!
    He said "automated process", aka an algorithm. There exist rings where there does not exist an algorithm which can compute if they are isomorphic or not.
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  5. #5
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    Re: Computing an isomorphism

    My problem needs only finite rings, although no matter how much computing power there is, brute-force stuff is out of the question. What if I take this to a lower level (although I think I've overrated the stuff to rings). Let A and B be to sets with Card(A)<= Card(B). Can there be found an isomorphism between those two sets? If yes what the conditions must be respected by the sets (being finite is one of them). Or another way of putting this:
    let A,B to fnitie sets . I know A and I know B
    I need to know:

    If there is a function f:A->B (or a know restriction of B) that can find an application of A in B, and I need that function to be injective and surjective.
    Last edited by jazzmoke; August 25th 2011 at 06:49 AM.
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