Rings of fractions
Rings of fractions, and the associated localization process, correspond to the algebrogeometric picture of studying behavior of an open set or near a point. Using the way we construct
but this definition only works if
Let
We also have the ring homomorphism
Let
The ring

is unit in . 
for some . 
Every element of
has the form for some .
Conversely, these properties determine
Some examples follow:

Let
be a prime ideal in . Then is multiplicatively closed and is written as , the localization of at . with form an ideal in . If , and , then is unit. is the only maximal ideal in , and is a local ring. If is the set of prime numbers, then is the set of all rationals with denominators as powers of . 
is the zero ring . 
Let
. Then we write for in this case. Let be a polynomial ring where is a field, and a prime ideal in . Then, is the ring of all rational functions . Let be a variety defined by , that is to say the set such that whenever . Then, can be identified with the ring of all rational functions in which are defined at almost all points of ; it is the local ring of defined along variety . This is the prototype of local rings, as they arise in algebraic geometry. 
Let
be an ideal in , such that . Then, is multiplicatively closed.
Modules of fractions
The same construction involving
then, this sequence is exact at
This can proved by noticing that if
If
We have the following properties of fractions of modules, given that

. 
. 
.
The first is obvious. To prove (ii), consider
To prove (iii), apply
Let
To prove this, notice that the mapping
Hence,
If
In particular, if
Local properties
A ring

. 
, for all prime ideals of . 
, for all maximal ideals of .
Since (i)
Let

is injective. 
is injective for each prime ideal . 
is injective for each maximal ideal .
Flatness is a local property. For any

is a flat module. 
is flat for each prime ideal . 
is flat for each maximal ideal .
Primary decomposition
Just as prime ideals are generalizations of primes in
In other words, for
Clearly, every prime ideal is primary. Also, the contraction of a primary ideal is primary, for if
Let
Examples:

and are primary in , where is a prime number. 
Let
and . Then, , in which all zerodivisors are multiples of , hence nilpotent. Hence, is primary and its radical is such that , so that a primary ideal is not necessarily a prime power. 
Let
. is not primary because , but and .
If
If
Let

If
, then the ideal quotient . 
If
then is primary, and hence . 
If
, then .
Let us prove (iii), since (i) and (ii) are obvious. Let
The primary decomposition of ideal
An ideal is said to be decomposable if such a decomposition exists.
First uniqueness theorem: Let
Let
Let
In particular, if the zero ideal is decomposable, then the set
Let

If
, then . 
If
, then is primary, and its contraction in is .
Let
Moreover, these are minimal primary decompositions.
The set
Second uniqueness theorem: Let
Integral dependence
Let
Let
the
The following are equivalent:

is integral over . 
is a finitely generated module. 
is contained in a subring of , such that is a finitely generated module. 
There exists a faithful
module which is finitely generated as an module.
To prove (i)
for all
C is termed nonsingular if
The goingup theorem
The goingdown theorem
Valuations
Chain conditions
Noetherian rings
In a Noetherian ring
Artin rings
Discrete valuation rings
By convention,
Valuation ring of a discrete valuation
For valuation ring
The Nullstellansatz
Variety
Tautologically,
The Nullstellansatz states that:

If
then 
, the radical of the ideal
A variety is irreducible if it cannot be expressed as the union of two proper subvarieties:
The following reverseinclusions are obvious:
Zariski topology
Zariski topology is a topology where the only closed sets are the algebraic ones, the zeros of polynomials. The Zariski topology on a variety is Noetherian.
corresponds exactly to the Zariski topology on
where
Localization
For a general construction, let
Then, for ideal
These three statements are equivalent:

is local if it has a unique maximal ideal . 
is the unique maximal ideal. 
If
is a maximal ideal and , then is unit.
Support and annihilator
Support of M is defined as
Example: If

If
is generated by a single element , such that , then . 
If
and , then . 
If
and , then . 
If
, then .
In the disjoint union