# Is it true “If $X \subset Y$ then $\Bbb I(Y) \subset \Bbb I(X)$”(proper inclusion)?

Is it true "If $$X \subset Y$$ then $$\Bbb I(Y) \subset \Bbb I(X)$$; here I am using proper inclusion. Couldn't prove it though. Trying for long time please help. Actually I saw here "https://people.maths.bris.ac.uk/~mp12500/teaching/Lectures5-7.pdf" in proposition 17 $$\Rightarrow$$ direction.

Looks like it's actually Prop 11 (part 2) in the notes you link that shows inclusion.

To see proper inclusion, suppose $$I(Y) = I(X)$$. Then $$V(I(Y)) = V(I(X))$$. But as $$X$$ and $$Y$$ are algebraic varieties, $$V(I(X)) = X$$ and $$V(I(Y)) = Y$$. So you must have had $$X = Y$$.

• Would like to vote my question? – Gimgim Feb 11 at 21:56