Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R,[1] or a module of finite type.
Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.
A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.
Definition
The left R-module M is finitely generated if there exist a1, a2, ..., an in M such that for any x in M, there exist r1, r2, ..., rn in R with x = r1a1 + r2a2 + ... + rnan.
The set {a1, a2, ..., an} is referred to as a generating set of M in this case. A finite generating set need not be a basis, since it need not be linearly independent over R. What is true is: M is finitely generated if and only if there is a surjective R-linear map:
for some n (M is a quotient of a free module of finite rank).
If a set S generates a module that is finitely generated, then there is a finite generating set that is included in S, since only finitely many elements in S are needed to express the generators in any finite generating set, and these finitely many elements form a generating set. However, it may occur that S does not contain any finite generating set of minimal cardinality. For example the set of the prime numbers is a generating set of viewed as -module, and a generating set formed from prime numbers has at least two elements, while the singleton{1} is also a generating set.
In the case where the
Any module is the union of the directed set of its finitely generated submodules.
A module M is finitely generated if and only if any increasing chain Mi of submodules with union M stabilizes: i.e., there is some i such that Mi = M. This fact with
Examples
- If a module is generated by one element, it is called a cyclic module.
- Let R be an integral domain with K its field of fractions. Then every finitely generated R-submodule I of K is a fractional ideal: that is, there is some nonzero r in R such that rI is contained in R. Indeed, one can take r to be the product of the denominators of the generators of I. If R is Noetherian, then every fractional ideal arises in this way.
- Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups. These are completely classified by the structure theorem, taking Z as the principal ideal domain.
- Finitely generated (say left) modules over a division ring are precisely finite dimensional vector spaces (over the division ring).
Some facts
Every
In general, a module is said to be Noetherian if every submodule is finitely generated. A finitely generated module over a Noetherian ring is a Noetherian module (and indeed this property characterizes Noetherian rings): A module over a Noetherian ring is finitely generated if and only if it is a Noetherian module. This resembles, but is not exactly Hilbert's basis theorem, which states that the polynomial ring R[X] over a Noetherian ring R is Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring.
More generally, an algebra (e.g., ring) that is a finitely generated module is a finitely generated algebra. Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (See integral element for more.)
Let 0 → M′ → M → M′′ → 0 be an exact sequence of modules. Then M is finitely generated if M′, M′′ are finitely generated. There are some partial converses to this. If M is finitely generated and M′′ is finitely presented (which is stronger than finitely generated; see below), then M′ is finitely generated. Also, M is Noetherian (resp. Artinian) if and only if M′, M′′ are Noetherian (resp. Artinian).
Let B be a ring and A its subring such that B is a
Finitely generated modules over a commutative ring
For finitely generated modules over a commutative ring R,
Any R-module is an
An example of a link between finite generation and integral elements can be found in commutative algebras. To say that a commutative algebra A is a finitely generated ring over R means that there exists a set of elements G = {x1, ..., xn} of A such that the smallest subring of A containing G and R is A itself. Because the ring product may be used to combine elements, more than just R-linear combinations of elements of G are generated. For example, a polynomial ring R[x] is finitely generated by {1, x} as a ring, but not as a module. If A is a commutative algebra (with unity) over R, then the following two statements are equivalent:[5]
- A is a finitely generated R module.
- A is both a finitely generated ring over R and an integral extension of R.
Generic rank
Let M be a finitely generated module over an integral domain A with the field of fractions K. Then the dimension is called the generic rank of M over A. This number is the same as the number of maximal A-linearly independent vectors in M or equivalently the rank of a maximal free submodule of M (cf. Rank of an abelian group). Since , is a
Now suppose the integral domain A is generated as algebra over a field k by finitely many homogeneous elements of degrees . Suppose M is graded as well and let be the Poincaré series of M. By the
A finitely generated module over a principal ideal domain is torsion-free if and only if it is free. This is a consequence of the structure theorem for finitely generated modules over a principal ideal domain, the basic form of which says a finitely generated module over a PID is a direct sum of a torsion module and a free module. But it can also be shown directly as follows: let M be a torsion-free finitely generated module over a PID A and F a maximal free submodule. Let f be in A such that . Then is free since it is a submodule of a free module and A is a PID. But now is an isomorphism since M is torsion-free.
By the same argument as above, a finitely generated module over a
Equivalent definitions and finitely cogenerated modules
The following conditions are equivalent to M being finitely generated (f.g.):
- For any family of submodules {Ni | i ∈ I} in M, if , then for some finite subset F of I.
- For any chain of submodules {Ni | i ∈ I} in M, if , then Ni = M for some i in I.
- If is an epimorphism, then the restriction is an epimorphism for some finite subset F of I.
From these conditions it is easy to see that being finitely generated is a property preserved by Morita equivalence. The conditions are also convenient to define a dual notion of a finitely cogenerated module M. The following conditions are equivalent to a module being finitely cogenerated (f.cog.):
- For any family of submodules {Ni | i ∈ I} in M, if , then for some finite subset F of I.
- For any chain of submodules {Ni | i ∈ I} in M, if , then Ni = {0} for some i in I.
- If is a monomorphism, where each is an R module, then is a monomorphism for some finite subset F of I.
Both f.g. modules and f.cog. modules have interesting relationships to Noetherian and Artinian modules, and the Jacobson radical J(M) and socle soc(M) of a module. The following facts illustrate the duality between the two conditions. For a module M:
- M is Noetherian if and only if every submodule N of M is f.g.
- M is Artinian if and only if every quotient module M/N is f.cog.
- M is f.g. if and only if J(M) is a superfluous submoduleof M, and M/J(M) is f.g.
- M is f.cog. if and only if soc(M) is an essential submoduleof M, and soc(M) is f.g.
- If M is a semisimple module (such as soc(N) for any module N), it is f.g. if and only if f.cog.
- If M is f.g. and nonzero, then M has a maximal submoduleand any quotient module M/N is f.g.
- If M is f.cog. and nonzero, then M has a minimal submodule, and any submodule N of M is f.cog.
- If N and M/N are f.g. then so is M. The same is true if "f.g." is replaced with "f.cog."
Finitely cogenerated modules must have finite
Another formulation is this: a finitely generated module M is one for which there is an epimorphism mapping Rk onto M :
- f : Rk → M.
Suppose now there is an epimorphism,
- φ : F → M.
for a module M and free module F.
- If the kernel of φ is finitely generated, then M is called a finitely related module. Since M is isomorphic to F/ker(φ), this basically expresses that M is obtained by taking a free module and introducing finitely many relations within F (the generators of ker(φ)).
- If the kernel of φ is finitely generated and F has finite rank (i.e. F = Rk), then M is said to be a finitely presented module. Here, M is specified using finitely many generators (the images of the k generators of F = Rk) and finitely many relations (the generators of ker(φ)). See also: free presentation. Finitely presented modules can be characterized by an abstract property within the category of R-modules: they are precisely the compact objects in this category.
- A coherent module M is a finitely generated module whose finitely generated submodules are finitely presented.
Over any ring R, coherent modules are finitely presented, and finitely presented modules are both finitely generated and finitely related. For a Noetherian ring R, finitely generated, finitely presented, and coherent are equivalent conditions on a module.
Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and a finitely related flat module is projective.
It is true also that the following conditions are equivalent for a ring R:
- R is a right coherent ring.
- The module RR is a coherent module.
- Every finitely presented right R module is coherent.
Although coherence seems like a more cumbersome condition than finitely generated or finitely presented, it is nicer than them since the category of coherent modules is an abelian category, while, in general, neither finitely generated nor finitely presented modules form an abelian category.
See also
References
- ^ For example, Matsumura uses this terminology.
- ^ Bourbaki 1998, Ch 1, §3, no. 6, Proposition 11.
- ^ Matsumura 1989, Theorem 2.4.
- ^ Atiyah & Macdonald 1969, Exercise 6.1.
- ^ Kaplansky 1970, p. 11, Theorem 17.
- ^ Springer 1977, Theorem 2.5.6.
Textbooks
- Atiyah, M. F.; Macdonald, I. G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., pp. ix+128, MR 0242802
- ISBN 3-540-64239-0
- Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021
- Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Springer-Verlag, ISBN 978-0-387-98428-5
- ISBN 978-0-201-55540-0
- Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Translated from the Japanese by M. Reid (2 ed.), Cambridge: Cambridge University Press, pp. xiv+320, MR 1011461
- Springer, Tonny A. (1977), Invariant theory, Lecture Notes in Mathematics, vol. 585, Springer, ISBN 978-3-540-08242-2.