Finitely-generated module

In mathematics, a module is a finitely-generated module if it has a finite generating set.

Contents

1 Intuitive introduction 2 Formal definition 3 Some facts 4 Finitely-presented module

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.: 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.

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). 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. 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}. Another one would be {2,1/6}.

Formal definition

The left R-module M is finitely-generated if and only if there exist a₁, a₂, ..., a_n in M such that for all x in M, there exist r₁, r₂, ..., r_n in R with x = r₁a₁ + r₂a₂ + ... + r_na_n.

The set {a₁, a₂, ..., 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).

Some 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.

Every homomorphic image of a finitely generated module is finitely generated. In general, submodules of a finitely generated modules need not be finitely generated. (As an example, consider the ring R=Z[X₁,X₂,...] 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 module

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)).

For a noetherian ring R this condition is automatic; so in many common cases there is no distinction between finitely-generated module and finitely-presented module. However, for a general ring R, it is an important concept, as is evidenced by the fact that the category of all finitely presented R-modules is abelian, something that's not generally true for the category of all finitely-generated R-modules.