What Are the Ideals and Units in These Factor Rings?

[logarithmic]

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?

 

[Office_Shredder]

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.

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?

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?

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