# Generating set of a module

In algebra, a **generating set** *G* of a module *M* over a ring *R* is a subset of *M* such that the smallest submodule of *M* containing *G* is *M* itself (the smallest submodule containing a subset is the intersection of all submodules containing the set). The set *G* is then said to generate *M*. For example, the ring *R* is generated by the identity element 1 as a left *R*-module over itself. If there is a finite generating set, then a module is said to be finitely generated.

Explicitly, if *G* is a generating set of a module *M*, then every element of *M* is a (finite) *R*-linear combination of some elements of *G*; i.e., for each *x* in *M*, there are *r*_{1}, ..., *r*_{m} in *R* and *g*_{1}, ..., *g*_{m} in *G* such that

Put in another way, there is a surjection

where we wrote *r*_{g} for an element in the *g*-th component of the direct sum. (Coincidentally, since a generating set always exists; for example, *M* itself, this shows that a module is a quotient of a free module, a useful fact.)

A generating set of a module is said to be **minimal** if no proper subset of the set generates the module. If *R* is a field, then it is the same thing as a basis. Unless the module is finitely-generated, there may exist no minimal generating set.[1]

The cardinality of a minimal generating set need not be an invariant of the module; **Z** is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set { 2, 3 }. What is uniquely determined by a module is the infimum of the numbers of the generators of the module.

Let *R* be a local ring with maximal ideal *m* and residue field *k* and *M* finitely generated module. Then Nakayama's lemma says that *M* has a minimal generating set whose cardinality is . If *M* is flat, then this minimal generating set is linearly independent (so *M* is free). See also: minimal resolution.

A more refined information is obtained if one considers the relations between the generators; cf. free presentation of a module.

## References

- Dummit, David; Foote, Richard.
*Abstract Algebra*.