Definition
A free module is a module with a basis: a linearly independent generating set.
For an -module, the set is a basis for if:
- is a generating set for ; that is to say, every element of is a finite sum of elements of multiplied by coefficients in ;
- is linearly independent, that is, if for distinct elements of, then (where is the zero element of and is the zero element of ).
If has invariant basis number, then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the rank of the free module, and is said to be free of rank n, or simply free of finite rank if the cardinality is finite.
Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each .
The definition of an infinite free basis is similar, except that will have infinitely many elements. However the sum must still be finite, and thus for any particular only finitely many of the elements of are involved.
In the case of an infinite basis, the rank of is the cardinality of .
Read more about this topic: Free Module
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)