Algebraic set defined by element in coordinate ring

Question

Given an algebraic set $Y=Z(T)$ and consider $\overline{f}\in A(Y)$. Can we safely conclude that $$Z(\overline{f})=\{P\in\mathbb{A}^n_k| f(P)=0,\forall f\in\overline{f}\}=Z(f)\cap Z(T)$$ I think we can because the set $\overline{f}$ contains all possibilies of $f+g$ where $g\in (T)$. Hence $Z(f+g)=Z(f)\cap Z(g)$. Is there any mistakes in the argument?
This question comes from my attempt in solving the exercise 1.1.8 from Hartshorne. Thank you!