**Restatement**: An affine algebraic variety, the set of common zeros of a collection of polynomials, is an irreducible closed subset of (with the induced topology). An open subset of an affine variety is called a quasi affine algebraic variety.

In this post examples affine and quasi-affine object varieties will be given after an analysis of the relationship between the Z(*) and I(*) operations are clarified.

**Definition 2.1** The radical of any ideal, , is defined as

An ideal that is equal to its radical is called a radical ideal, i.e. if .

**Example 2.1** The radical of the ideal , integer mutilples of , is . The radical ideal of is .

**Definition 2.2** Hilbert’s Nullstellensatz. If is an algebraically closed field with an ideal in and a polynomial that vanishes at all points of , then for some integer .

**Proposition 2.1** Properties of the operations Z(*) and I(*).

1) If are subsets of , then .

2) If are subsets of , then .

3) For any two subsets , of , we have .

4) For any ideal , . This is a consequence of Definition 2.2.

5) For any subset , , the closure of .

**Definition 2.3** Prime ideals. An ideal of a commutative ring is prime if it has the following properties:

1) if such that , then or ,

2) is not equal to .

**Definition 2.4** Maximal ideals. is a maximal ideal of a ring if there are no other ideals contained between and . Every maximal ideal is prime.

**Example 2.2** In the ring of integers, , the maximal ideals are the principle ideals generated by a prime number.

**Corollary 2.1** If is algebraically closed

1) There is a correspondence .

2) Every ideal not a subset of has a zero in .

**Example 2.3** Let be a finite algebraic set. is generated by the function , and so is the inverse of .

**Corollary 2.2** An algebraic set is an affine variety iff its ideal is a prime ideal.

**Example 2.4** has a dimension of as single points are the only irreducible closed subsets of not equal to . In fact has a dimension of .

**Example 2.5** If is an irreducible polynomial in , the affine variety is called a surface if and a hyper-surface of .

Any finite topological space can be decomposed uniquely into finitely many irreducible spaces.

**Definition 2.5** A topological space is said to be Noetherian if it satisfies a descending chain condition for closed subsets. For any sequence of closed subsets there is an integer such that . Another way of saying this is that the closed subsets are stationary.

As is a Noetherian ring this implies that any algebraic set is a Noetherian topological space.

**Definition 2.6** Inclusion-reversing is defined for two sets and with a mapping . Then is inclusion-reversing iff for every pair of sets such that .

**Corollary 2.3** There is a one-to-one inclusion-reversing correspondence between algebraic sets in and radical ideals in , given by and .

To summarize the following correspondences between geometric and algebraic forms have been established:

1) points in affine space maximum ideals in a polynomial ring. (Corollary 2.1)

2) affine varieties prime ideals. (Corollary 2.2)

3) algebraic sets radical ideals. (Corollary 2.3)