Hello! I want to ask: what should I know so that I can answer this: Is the ideal I=(X+Y,X−Y) in the polynomial ring C[X,Y] a prime ideal?
To answer this, I have the following two ideas, are these enough to answer the above question?:
1) One way to solve this of course would be to solve it by definition i.e., taking f(x,y) and g(x,y) from C[x,y] and then noting that there exist p(x,y) and q(x,y) in C[x,y] such that f(x,y)g(x,y)=p(x,y)(x+y)+q(x,y)(x-y) and then concluding from here that either p(x,y) or q(x,y) is in (x+y,x-y). But this seems very lengthy too me.
To answer this, I have the following two ideas, are these enough to answer the above question?:
1) One way to solve this of course would be to solve it by definition i.e., taking f(x,y) and g(x,y) from C[x,y] and then noting that there exist p(x,y) and q(x,y) in C[x,y] such that f(x,y)g(x,y)=p(x,y)(x+y)+q(x,y)(x-y) and then concluding from here that either p(x,y) or q(x,y) is in (x+y,x-y). But this seems very lengthy too me.
Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal? Justify your answer.
I am a bit hesitant about asking this here. The question is not "How to Solve This Problem". The question is "What do I need to learn first in order to solve this type of problems".
I have ...
I wanted to check if my approach is correct:
Consider : $$\phi:\mathbb{C}[x, y] \rightarrow \mathbb{C}[x]$$ $$ y \mapsto -x$$ $$x \mapsto x$$ is a surjective homomorphism, with $\ker(\phi)=(x+y)$. Hence, $\mathbb{C}[x, y]/(x+y) \cong \mathbb{C}[x]$. Now, I use the third isomorphism theorem to as...
Could someone please give me a hint how to prove (preferably directly, without finding clever homomorphisms) that $(X,Y)$ is a maximal ideal in $\mathbb{C}[X,Y]$ ?
This ideal is prime, since it contains all polynomials of all degree that don't have a constant term, so my guess was, it's maybe al...
