Inmathematics, agenerating set Γ of amoduleM over aringR is asubset ofM such that the smallestsubmodule ofM containing Γ isM itself (the smallest submodule containing a subset is theintersection of all submodules containing the set). The set Γ is then said to generateM. For example, the ringR is generated by the identity element 1 as a leftR-module over itself. If there is afinite generating set, then a module is said to befinitely generated.

This applies toideals, which are the submodules of the ring itself. In particular, aprincipal ideal is an ideal that has a generating set consisting of a single element.

Explicitly, if Γ is a generating set of a moduleM, then every element ofM is a (finite)R-linear combination of some elements of Γ; i.e., for eachx inM, there arer₁, ...,r_m inR andg₁, ...,g_m in Γ such that

${\displaystyle x=r_1{g}_1+\cdots +r_m{g}_m.}$

Put in another way, there is asurjection

${\displaystyle \underset{g\in \mathrm{\Gamma }}{⨁}R\to M,r_g\mapsto r_{g}g,}$

where we wroter_g for an element in theg-th component of the direct sum. (Coincidentally, since a generating set always exists, e.g.M itself, this shows that a module is aquotient of afree module, a useful fact.)

A generating set of a module is said to beminimal if noproper subset of the set generates the module. IfR is afield, then a minimal generating set is the same thing as abasis. Unless the module isfinitely generated, there may exist no minimal generating set.^[1]

Thecardinality 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}. Whatis uniquely determined by a module is theinfimum of the numbers of the generators of the module.

LetR be alocal ring withmaximal idealm andresidue fieldk andM finitely generated module. ThenNakayama's lemma says thatM has a minimal generating set whose cardinality is${\displaystyle {\dim }_{k}M/mM={\dim }_{k}M{\otimes }_{R}k}$. IfM isflat, then this minimal generating set islinearly independent (soM is free). See also:Minimal resolution.

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

• Countably generated module
• Flat module
• Invariant basis number

• Dummit, David; Foote, Richard.Abstract Algebra.

• Abstract algebra

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