Let $$ P_1 = (0, 0), \quad P_2 = (1, 0), \quad P_3 = (0, 1), \quad P_4 = (1, 1). $$ Let $Y = \{P_1, P_2, P_3, P_4\}$ and let $I \subset k[x, y]$ be the ideal of $Y$ .
Show that the affine coordinate ring $A(Y) = k[x, y]/I$ of Y has dimension $4$ as $k$-vector space.
Show that $I = (f, g)$, where $f = x^2 − x$ and $g = y^2 − y$.
Hint: show that $(f, g) \subseteq I$, then that $(1, x, y, xy)$ gives a k-basis for $k[x, y]/(f, g)$ using divisions with remainder, then that the natural morphism $k[x, y]/(f, g) \to A(Y)$ is an isomorphism.
For question 1 I think that a $k$-algebra morphism $$ k[x, y] \to k^4, \qquad f \mapsto (f(P_1), f(P_2), f(P_3), f(P_4)) $$ will do the job but I am really lost with the structure of the spaces to form it.
Can anyone offer some help?
Well, the coordinate ring is $k[x,y]/J$, where $J$ is the ideal $$J=\langle x,y\rangle\cap\langle x-1,y\rangle \cap\langle x,y-1\rangle\cap \langle x-1,y-1\rangle,$$ which is equal to $k[x,y]/\langle x^2-x,y^2-y\rangle$. See Showing that two ideals are equivalent. for more details.
Moreover, the standard monomial basis of $k[x,y]/J$ consists of $1,x,y,xy$ and so the coordinate ring is 4-dimensional over $k$.