**<The Detail of This Chapter.pdf>**

a fixed algebraically closed field k

a n dimensions space **A** which's coordinates is denoted with k

a kind of functions from **A **to k, which construct a polynomial
ring R=F[x_{1},_{
}..., x_{n}] over k

to a polynomial, we have its zeros (solutions of a polynomial equation) which belong to **A**

to some polynomials, we have their *
zero set*
Z which's elements are common zeros of all these polynomials, which construct a
subset T of R.

as a ring, we can get an ideal a of R that is generated by its subset T

the zero set of T = the zero set of a

a's generators is finite

the zero set of T = the common zeros of the finite set of a's generators (polynomials)

for any subset T of R, which's zero set is a *
algebraic set* of space **A**

we denote the function that maps the subset T of R to the subset Y of** A**:
Y=Z(T)

conversely, for any subset Y of **A**, we can get a subset T of R, its
zero set cover Y, and the ideal that is generated by T is denoted with
I(Y), its zero set cover Y too, which is a function
which maps subsets of **A** to ideals of R.

then, we have two functions: Z(T) and I(Y), about the relationship about: T, a, Y, Z(T), I(Y)

proposition:

the complements of such algebraic sets endue **A **with a topology, name
it the *Zariski topology* on **A**

for more appropriate with polynomial ring, we set affine structure on **A**

to base on a Zariski topology, an irreducible closed subset of **A **is
named an *affine algebraic variety*, and the
correspondence open subset of an affine algebraic variety is named a *
quasi-affine algebraic variety*