We will assume all rings are commutative and unital.
Definition 1. Let be a ring, a right -module, and a left -module. is -flat if for every left -module and monomorphism the homomorphism is injective.
Some basic results can be shown:
Proposition 2. If is a right -module and is a left -module such that is -flat, then if we have that is -flat and -flat. Also if is a family of left -modules such that is -flat for all then is -flat.
Proposition 3. The following are equivalent for a right -module
- The canonical map is injective (where for some multiplicative subset ).
- is -flat for every left -module
- For every exact sequence of left -modules and homomorphisms
Definition 4. A right -module satisfying the equivalent conditions above is flat. (A dual definition exists for left modules)
Proposition 5. Let be a right -module, then the following are equivalent.
1. The sequence of left -modules is exact iff
2. is flat and for a left -module
3. is flat and for every homomorphism
4. is flat and for every maximal left ideal of
Definition 6. A right -module satisfying the above conditions is faithfully flat.
Recall that for a subset of an -module the annihilator of is defined as
An -module is faithful if
Definition 7. Let be a left -module and suppose
is an exact sequence such that each is free and finitely generated. Then is said to be finitely presented, and the exact sequence is called a finite presentation of of length
Note that if is a ring homomorphism, then a finite presentation of a left -module induces a finite presentation of where the th term of the first sequence, becomes with (We will consider to be the term of the original sequence)
Proposition 8. Let be a left -module.
- If admits a finite presentation, then is finitely generated.
- If is left Noetherian, then if is finitely generated, it admits a finite presentation.
- If is projective and finitely generated, then it admits a finite presentation.
Definition 9. Let be a left -module. is pseudo-coherent if every finitely generated submodule of is admits a finite presentation. is coherent if it is pseudo-coherent and finitely generated.
 Bourbaki, N. Commutative Algebra, Ch 1-7. Springer-Verlag. 1989.