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The cardinality of a basis of a free module.
Let be a free module. The rank of is the cardinality of a basis, denoted . This is well-defined: any two bases of a free module have the same cardinality.
When the rank is finite, it plays the role of dimension, but over a general ring one typically speaks of rank only for free (or locally free) modules.
A left -module is free if it has a basis , i.e. every element of admits a unique expression as a finite -linear combination of elements of .
Free modules are characterized by the universal property of free modules: functions from a basis extend uniquely to module homomorphisms. Over a field, free modules coincide with vector spaces and their bases agree with linear-algebra bases.
A cardinality is the size of a set, understood up to bijection: two sets and have the same cardinality if there exists a bijective function . The cardinality of is often denoted .
Let be a free -module. A basis of is a subset such that every can be written uniquely as a finite sum
with coefficients , where all but finitely many are zero. Uniqueness is equivalent to the familiar notions of linear independence and spanning (compare linear combinations in linear algebra).
Bases generalize the concept of a basis in a vector space, but over rings bases can fail to exist even for finitely generated modules.
Let be a vector space over , and let .
A set is a basis (also called a Hamel basis) of if:
If has a basis with finitely many elements, then is finite-dimensional, and the number of basis vectors is the dimension . If no finite basis exists, is infinite-dimensional (often written ).