Trivial (?) alg. geometry problem

[Kalidor]

Trivial (!?) alg. geometry problem

Homework Statement

Consider $$Y=Q_1,Q_2,\ldots,Q_r \subset \mathbb{A}^n$$, a finite set of $$r$$ different points. What are the generators of the ideal $$I(Y)$$

The Attempt at a Solution

Knowing that $$I(Q_i)=(X_1-Q_{i,1},\ldots,X_n-Q_{i,n})$$ and so on, my guess would be that the solution is something like
$$(\prod_{k=1}^r f_{k,i}), 1 \leq i \leq n$$ with $$f_{k,i} \in I(Q_k)$$

It seems kind of messed. Any ideas?

 

[Kalidor]

My notation is too messy?