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In mathematics, a generating set Γ of a module M over a ring R is a subset of M such that the smallest submodule of M containing Γ is M itself. The set Γ 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.
This applies to ideals, which are the submodules of the ring itself. In particular, a principal ideal is an ideal that has a generating set consisting of a single element.
Explicitly, if Γ is a generating set of a module M, then every element of M is a R-linear combination of some elements of Γ; i.e., for each x in M, there are r1,..., rm in R and g1,..., gm in Γ such that
Put in another way, there is a surjection