Definition 1.15 (Polynomial functions and coordinate rings). Let $X \subset \mathbb{A}^n$ be an affine variety. A polynomial function on $X$ is a map $X \to K$ that is of the form $x \mapsto f(x)$ for some $f \in K[x_1,\ldots,x_n]$. By Remark 1.14 (b) the ring of all polynomial functions on $X$ is just the quotient ring
$$A(x) := k[x_1, \ldots, x_n] \big / I(X).$$
It is called the coordinate ring of the affine variety $X$.
If Y is embedded via the inclusion maps in n-affine space and some m-affine space as well then how can we show that the corresponding coordinate rings of Y are isomorphic as k-algebras?
The reason I ask this is that I have been told that the coordinate rings really capture the geometry of the affine variety.The embedding of an affine variety in a bigger space does not matter because you get an isomorphism as k-algebras of the corresponding coordinate rings.So in that sense being an affine variety is a geometrically intrinsic property.
The definition above can be found in Gathmann's notes page-10 Definition 1.15 here: Notes on Algebraic Geometry