Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal?

Question

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 little to no grip on the topic of Rings & Fields, let alone Prime Ideals.

What does $\mathbb{C}[X,Y]$ mean here? A polynomial in two variables $X$ and $Y$, whose coefficients are picked from the set of complex numbers? A google search tells that Prime Ideals share many resemblances with prime numbers. Now, (intuitively) an Ideal is a special subset of a ring, that can sustain multiplication by the elements of the ring (from left or from right. If commutative, then distinction doesn't matter). Now, how does this apply to $I=(X+Y,X-Y)$?

Also, it will be very helpful if someone tells me how to build up my strength in this particular area of Mathematics (I have a moderately well understanding of group theory; although not at all deep) .

This question is indeed very unclear and pretty much opinion based and unsuited for this site. I am myself very much confused about this. Any help is appreciated.

Answer 1

I do think this question should be answered since there are out there many people trying to learn math as you are, a little bit of help can't hurt right? Also this question has now 5 upvotes, meaning someone is actually interested in it.

I will try to answer all your question marks.

1. Yes it is a prime Ideal. A prime ideal is an ideal which the kernel of a ring morphism to a domain. You see that you may define a ring morphism from $\mathbb{C}[x,Y]$ to $\mathbb{C}$ by sending both $X$ and $Y$ to $0$, and fixing all complex numbers. This has kernel $(X,Y)$, which is the same as $I=(X+Y,X-Y)$ since $2X=X+Y+(X-Y)$ and $-2Y=X+Y-(X-Y)$.
2. $\mathbb{C}[X,Y]$ is the set of all polynomials in $X$ and $Y$ with complex coefficients. It is both a complex vector space and a ring. These two structures are compatible, meaning it is a $\mathbb{C}$-algebra.
3. As by point 1, yes it is. Aside about the google search. When you will have read the proper definition, you may want to study the prime ideals of the ring $\mathbb{Z}$ and see how they prime ideals are connected with prime numbers.
4. Rather than "apply", we understand that your set $I$, which is the set generated by the two elements $X+Y$ and $X-Y$, satisfies this definion, and hence it satisfies any property of an ideal. (For example, it is the kernel of a morphism).

As a final remark, I would like to reccomend you the brilliant Dummit and Foote of abstract algebra, which covers all this material and much more in a clear and example-based way.