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

the sequence

is exact.

**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

is exact.

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.

[1] Bourbaki, N. *Commutative Algebra, Ch 1-7*. Springer-Verlag. 1989.