Let $A=k[x,y]/(xy,y^2)$. Support of $f\in A$ is defined as $supp(f)=\{p\in Spec(A):f\neq 0\in A_p\}$. What are all possible supports of $f\in A$ ?
$Spec A$ is simply a line $k[x]$ with nonreduced stalk over $(x)$. It's easy to show that: $supp(f)=\{p\in Spec(A): ann(f)\subset p\}$. Using this I could find the following: $$supp(0)=\emptyset$$ $$supp(yf)=(x,y) \text{ such that } x\nmid f \text{ and} y\nmid f $$ $$supp(1)=Spec(A)$$
How can I find support for arbitrary $f$ ?
Hint: since you modded out by $xy$, any polynomial can be decomposed in to $f(x)+g(y)+c$ where $c\in k$ and $f,g$ have no constant terms. What happens to each of these terms when you go to stalks?