Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set. Equivalently, it is a homomorphic image of a free module on finitely many generators. The kernel of this homomorphism need not be finitely generated (then the module is finitely presented); over a Noetherian ring both concepts coincide. A finitely generated module over a field is simply a finite-dimensional vector space.

Intuitive introduction

Informally, modules are an abstraction of the concept of a number of directions, together with distances (or coefficients) in each direction. A generating set is a list which spans all the possible directions. A finitely generated module is one for which there is a finite generating set.

This image should nonetheless be used with care, because in a given module "distance" might not be interpreted as a continuous quantity (see examples 2 and 3 below of modules where "distance" is always a whole number). In some modules counter-intuitive things might happen if you travel far enough in one direction (for example in some modules you will get back to where you started). See also torsion modules.

Example 1.: The integer lattice, $Z^2$, can be viewed as a module over the integers under addition (as any abelian group). To informally interpret this example, consider ordinary map co-ordinates, East-West and North-South. Only two directions are required to span the whole map. Ignoring obstructions, you could get to any point on the map by travelling some distance East-West and then some other distance North-South. Thus we say that the whole area of the map is generated by the set {1 mile east, 1 mile north} together with coefficients from the real numbers. The map can be described as a finitely generated module (in fact, a 2-generator module) -- although for technical reasons it has to go as far as you like in all directions.

More generally, any ring is a module with a single generator over itself. Thus the ring of real polynomials $R\; [x]$ is finitely generated over itself, while it is an infinite-dimensional vector space over the reals. The direct sum of finitely many copies of a ring is finitely generated over this ring, like $Z^2$ in the above example - the result is a free module. Any finitely generated module can be obtained as a quotient module of a free module over some (not necessarily finitely generated) submodule.

Example 2. (not finitely generated module). Consider the positive rational numbers written as powers of prime numbers. So for example we express 18 as 2.3², 7/6 as 7.2^-1.3^-1 and so on. Here, the prime numbers are the "directions", and the exponent of each prime is the coefficient. When described in this way, the positive rationals form a module (over the integers). The abelian group structure in the module is the multiplication of rationals and multiplication by a ring element (an integer) corresponds to exponentiation. A finite generating set would be a finite set of rational numbers which could, by raising them to any integer power and multiplying them together, be used to express any rational number. No such set exists, because there are infinitely many prime numbers, and no finite set of rational numbers can generate them all. Hence "this is not" a finitely generated module.

Example 3. The module in example 2 contains many finitely generated submodules, constituting interesting examples. In fact, any finite set of generators provides an example. Take the positive rational numbers which (after simplification) contain only the primes 2 and 3. So for instance 6, 10/45=2/9 and 1/12 belong to this set. This is a module over the integers, which is also finitely generated. A set of generators is, for example, {2,3}. (Remember that it is a module over the integers in the sense of exponentiation!) Another one would be {2,1/6}.

Formal definition

The left "R"-module "M" is finitely generated if and only if there exist "a"₁, "a"₂, ..., "a"_"n" in "M" such that for all "x" in "M", there exist "r"₁, "r"₂, ..., "r"_"n" in "R" with "x" = "r"₁"a"₁ + "r"₂"a"₂ + ... + "r"_"n""a"_"n".

The set {"a"₁, "a"₂, ..., "a"_"n"} is referred to as a generating set for "M" in this case.

In the case where the module "M" is a vector space over a field "R", and the generating set is linearly independent, "n" is "well-defined" and is referred to as the dimension of "M" ("well-defined" means that any linearly independent generating set has "n" elements: this is the dimension theorem for vector spaces).

ome facts

Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups; these are completely classified. The same is true for the finitely generated modules over any principal ideal domain; see the structure theorem.

Every homomorphic image of a finitely generated module is finitely generated. In general, submodules of finitely generated modules need not be finitely generated. (As an example, consider the ring "R"=Z ["X"₁,"X"₂,...] of all polynomials in countably many variables. "R" itself is a finitely generated "R"-module [with {1} as generating set] . Consider the submodule "K" consisting of all those polynomials without constant term. Since every polynomial contains only finitely many variables, the "R"-module "K" is not finitely generated.) However, if the ring "R" is Noetherian, then every submodule of a finitely generated module is again finitely generated (and indeed this property characterizes Noetherian rings).

If "M" is a module which has a finitely generated submodule "K" such that the factor module "M"/"K" is finitely generated, then "M" itself is finitely generated.

Finitely presented and coherent modules

Another formulation is this: a finitely generated module "M" is one for which there is a surjective module homomorphism

:φ : "R^k" → "M".

A finitely presented module "M" is one for which the kernel of φ can also be taken to be finitely generated. If this is the case, we essentially have "M" specified using finitely many generators (the images of the "k" generators of "R^k") and finitely many relations (the generators of ker(φ)). A coherent module "M" is one that is finitely generated and such that the kernel of any map "R^k" → "M" (not necessarily surjective) is also finitely generated.

Over any ring "R", coherent modules are finitely presented, and finitely presented modules are finitely generated. For a noetherian ring "R", all three conditions are actually equivalent.

Although coherence seems like a more cumbersome condition than the other two, 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.