In
mathematics, a generating set Γ of a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
''M'' over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' is a
subset 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 ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
of ''M'' containing Γ is ''M'' itself (the smallest submodule containing a subset is the
intersection 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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
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
:
Put in another way, there is a
surjection
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
:
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
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a
free module, a useful fact.)
A generating set of a module is said to be minimal if no
proper subset
In mathematics, 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 ...
of the set generates the module. If ''R'' is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, then a minimal generating set is the same thing as a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
. Unless the module is
finitely generated, there may exist no minimal generating set.
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 . What ''is'' uniquely determined by a module is the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of the numbers of the generators of the module.
Let ''R'' be a
local ring In abstract algebra, more specifically 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 varieties or manifolds, or of algebraic n ...
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 c ...
''m'' and
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
''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
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
, then this minimal generating set is
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
(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
In algebra, a free presentation of a module ''M'' over a commutative ring ''R'' is an exact sequence of ''R''-modules:
:\bigoplus_ R \ \overset \to\ \bigoplus_ R \ \overset\to\ M \to 0.
Note the image under ''g'' of the standard basis generate ...
.
See also
*
Countably generated module In mathematics, a module over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes from Kaplansky's theorem (Kaplansky 1958), which states that a proj ...
*
Flat module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact se ...
*
Invariant basis number
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over ''R'' have a well-defined rank. In the case of fields, the IBN property becomes ...
References
*Dummit, David; Foote, Richard. ''Abstract Algebra''.
Abstract algebra
{{algebra-stub