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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Homework Help: Questions with factor rings

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