Formal Definition
The 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 with x = r1a1 + r2a2 + ... + rnan.
The set {a1, a2, ..., an} 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).
Read more about this topic: Finitely-generated Module
Famous quotes containing the words formal and/or definition:
“That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.”
—Fred Rogers (20th century)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)