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 interepreted as a continuous quantity (see examples 2 and 3 below of modules where "distance" is always a whole number), and also because 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 rational numbers written as powers of prime numbers. So for example we express 4 as 22, 15 as 3.5, 18 as 2.32, 1/6 as 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 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 rational numbers whose denominator is 1, 2, 3 or 6 (after simplification: 16/12=4/3, so it belongs to this set). This is a module over the integers, which is also finitely generated. A set of generators is, for example, {1/1,1/2,1/3,1/6}, but also {1/6} -which is, obviously, minimal-.
For M a left R-module, M is finitely-generated if and only if there exist a1, a2, ... , an in M such that for all x in M, there exist r1, r2, ..., rn in R such that x = r1a1 + r2a2 + ... + rnan.
{a1, a2, ... , an} is referred to as a generating set for M. Since it has n elements, we say that M is an n-generator module.
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).Formal definition