Whenever I decide to learn something – and especially when it’s learning for learning’s sake – I make sure to be meticulous with things. In particular, whenever I see propositions stated without proof, I break out the old pen and paper and start verifying.

The purpose of this post is to examine a few of the properties on page 661 of Dummit and Foote. Some of the background notation needed is discussed in this previous entry.

**Claim.** The following properties of the map are very easy exercises. Let and be subsets of .

(7) .

(9) If is any subset of , then , and if is any ideal, then .

(10) If is an affine algebraic set then , and if then , i.e. and .

*Proof.* (7) Note that . In particular, a polynomial vanishes on if and only if it vanishes on each disjoint component of separately, which happens if and only if it vanishes on the entirety of *and* on the entirety of .

(9) Suppose . Clearly, any polynomial which vanishes on is in , and because if and only if on , is certainly contained in the zero set of . Therefore, . Note that the inclusion doesn’t necessarily reverse since may vanish for some element .

Next, suppose that is an ideal of the ring and that . By definition, the locus is the collection of all points for which for all . Then, certainly, for all , , i.e., is an element in the ideal that vanishes on . Hence, implies that .

(10) First, suppose that is an ideal and that is an affine algebraic set. It suffices to show that by way of two-sided inclusion. To that end, let . Then is in the zero-set of the ideal , whereby it follows that for all . But for any satisfying for arbitrary , *and* by definition. Hence, .

Conversely, if , then for all . But all such functions disappear for all values by definition of . This means that the polynomials for which are precisely the functions which satisfy for all . Hence, itself must be an element of , whereby the equality is proved.

The other expression is proved similarly and is omitted for brevity. Therefore, as claimed,

and ,

from which it follows that the maps and are inverses of one another under the construction given here.