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Homework Help: Questions with factor rings

  1. Jun 17, 2010 #1
    The first question is to find the ideals of R[x]/<x^2 - x>. I can see that the elements of the factor ring are of the form p(x) + <x^2 - x>, where p(x) is in R[x], which can be simplified to q(x)(x^2 - x) + r(x) + <x^2 - x> = r(x) + <x^2 - x>, where r(x) is of degree 1 or 0.

    Now I'm pretty much stuck. Can we say anything more specific about r(x)? i.e. is it true that R[x]/<x^2 - x> = {ax + b + <x^2 - x> | a,b in R}? So now how do I find the ideals? It's easy to check if something's an ideal though.

    My other question is to find the units in R = C[x,y]/<xy - 1>. So after writing out some definitions, this reduces to finding polynomials p(x,y) and q(x,y) not in <xy - 1>, such that p(x,y)q(x,y) = 1 (I think). So any element of C is a unit of R, what else is there? There may be some theorem that help simplify something. Any ideas?
     
  2. jcsd
  3. Jun 17, 2010 #2

    Office_Shredder

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    That's right. Intuitively when you mod out by <x2-x> you're saying that x and x2 should be treated the same (because x2-x is now equal to zero). So given any polynomial, you can repeatedly apply this to reduce, for example x5 to x, or any other power of x.

    Is R here a generic ring, or the real numbers?

    You need p(x,y,)q(x,y)+<xy-1>=1+<xy-1> which is a bit different. For example, if p=x and q=y

    (x+<xy-1>)(y+<xy-1>)=(xy+<xy-1>)=(1+<xy-1>)

    so x and y are both units
     
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