mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a generating set Γ of a module ''M'' over a ring ''R'' is a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

of ''M'' such that the smallest

submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...

of ''M'' containing Γ is ''M'' itself (the smallest submodule containing a subset is the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of all submodules containing the set). 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 In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...

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 (finite) ''R''-linear combination of some elements of Γ; i.e., for each ''x'' in ''M'', there are ''r''₁, ..., ''r''_''m'' in ''R'' and ''g''₁, ..., ''g''_''m'' in Γ such that :

x = r_1 g_1 + \cdots + r_m g_m.

Put in another way, there is a

surjection In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

\bigoplus_ R \to M, \, r_g \mapsto r_g g,

where we wrote ''r''_''g'' for an element in the ''g''-th component of the direct sum. (Coincidentally, since a generating set always exists, e.g. ''M'' itself, this shows that a module is a quotient of a

free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...

, a useful fact.) A generating set of a module is said to be minimal if no

proper subset In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...

of the set generates the module. If ''R'' is a field, then a minimal generating set is the same thing as a basis. Unless the module is finitely generated, there may exist no minimal generating set. The

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

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 . What ''is'' uniquely determined by a module is the

infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...

of the numbers of the generators of the module. Let ''R'' be a

local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...

with

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...

''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

\dim_k M / mM = \dim_k M \otimes_R k

. If ''M'' is flat, then this minimal generating set is

linearly independent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...

(so ''M'' is free). See also: Minimal resolution. A more refined information is obtained if one considers the relations between the generators; see Free presentation of a module.

References

{{reflist *Dummit, David; Foote, Richard. ''Abstract Algebra''. Abstract algebra