The term ‘rank’ is used in many contexts to number levels within a hierarchy.
Let be a ring and a module over . If is a field, then is a vector space and we speak of the dimension of ; in the general case, we may speak of the rank:
A collection of elements of is called a basis of (over ) if for every there is a unique collection of elements of such that for all but finitely many and .
If has a basis it is called a free module (over ). For many examples of (the invariant basis number rings), the cardinality only depends on and not on the choice of basis. It is called the rank of over , notation: . In any case, is called the free module of rank . If is a finitely generated free module then the rank is a finite number.
All of the following are invariant basis rings (source: Wikipedia):
any nontrivial commutative ring ,
the group ring for any field (or nontrival commutative ring?) and any group,
any Noetherian ring.
Besides the trivial ring (over which any module is free with any set as basis), an example of a ring without invariant basis number is the ring of -dimensional square matrices (over any ring) in which each column has only finitely many nonzero entries (which allows multiplication to be defined). As a module over itself, this ring is free on any inhabited finite set, as may be shown by using the equation (applied to the columns).
Given a linear map, hence a homomorphism of modules, its rank is the rank of its image-module.
Often this is considered for the case that the linear map is represented by a matrix and one speaks of the rank of a matrix.
Let be a locally ringed space and a -module. Then its rank at a point is the vector space dimension of the fiber over the residue field .
If is of finite type, then the rank at can equivalently be defined as the minimal number of elements needed to generate the stalk as a -module (by Nakayama's lemma). In this case, the rank is a upper semicontinuous function .
In the internal language of the sheaf topos , the rank of can internally quite simply be defined as the minimal number of elements needed to generate (taken as an element of the suitably completed natural numbers, i.e. the poset of inhabited upper sets). Under the correspondence of internal inhabited upper sets in and upper semicontinuous functions (details at one-sided real number), this definition coincides with the usual one if is of finite type; see this MathOverflow question.
See also rank of a coherent sheaf.
As a simple special case of the above, a vector bundle is said to have rank if each fiber is a vector space of dimension .
Every pure set within the von Neumann hierarchy appears first at some level given by an ordinal number; this number is its hereditary rank.
We may define this rank explicitly (and recursively) as follows:
where is the supremum operation on ordinals (literally the union for von Neumann ordinals) and is the successor operation (which is for von Neumann ordinals).
Recall that a cardinal number is said to be regular if < whenever < and < for all .
A functor has rank for some regular cardinal if preserves -filtered colimits. has rank when it has rank for some regular cardinal . A monad has rank () when its underlying endofunctor does.
The properties of functors with rank are discussed in section 5.5 of Borceux (1994).
Last revised on April 4, 2023 at 08:22:14. See the history of this page for a list of all contributions to it.