What is the definition of $\mathbb{R}[x,y]$ and $\langle x , y\rangle$?

Question

Could someone please clarify the definition of polynomial ring in two variables $\mathbb{R}[x,y]$? Does it consist of all objects in form $a_0 + a_1 x^{k_{1x}}y^{k_{1y}} + a_2 x^{k_{2x}}y^{k_{2y}} + \cdots$? In addition, What would be the ideal $\langle x,y \rangle$ in this case? Is it the subring with polynomials in form $f(x,y)x + g(x,y)y$?

Answer 1

Yes, you are right for both. Another way of seeing $\langle x,y\rangle$ is to say that this is the kernel of the evaluation map at the point $(0,0)$ ; that is the set of polynomials that vanish at $(0,0)$.

Answer 2

If you know already the definition of the polynomial ring $R[x]$, then you can just take $R[x,y]=R[x][y]$ as a definition. If you take the general definition from, say, wikipedia, then you can prove the following result:

Proposition: There exists a unique isomorphism between the polynomial ring $R[x,y]$ and the ring $R[x][y]$ of polynomials in $y$ with coefficients from $R[x]$, which is identity on $R$, and which sends $x$ to $x$ and $y$ to $y$.

Concerning the ideal $\langle x,y\rangle$, we can say different things. For example, $I= \langle x,y \rangle\subset k[x,y]$ is not principal, see here.

Answer 3

$\mathbf R[x,y]$ consists of all finite sums of monomials in two variables $cx^i y^j$, $c\in\mathbf R$, $i,j\in \mathbf N$ (there is a more abstract and rigourous definition, which is valid for any commutative ring).

The ideal $\langle x,y\rangle$ consists of all polynomials with constant term $0$. It is not a subring with the usual conventions since it does not have an identity element.