then and if the dimension of is denoted by then: If dim is finite then If dim is infinite then Generalizations A vector space can be seen as a particular case of a matroid, and in the latter there is a well-defined notion of dimension. The length of a module and the rank of an abelian group both have several properties similar to the dimension of vector spaces. The Krull dimension of a commutative ring, named after Wolfgang Krull (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring. Trace The dimension of a vector space may alternatively be characterized as the trace of the identity operator. For instance, This appears to be a circular definition, but it allows useful generalizations. Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an algebra with maps (the inclusion of scalars, called the unit) and a map (corresponding to trace, called the counit). The composition is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in bialgebras, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension (), so in these cases the normalizing constant corresponds to dimension. Alternatively, it may be possible to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "trace class operators" on a Hilbert space, or more generally nuclear operators on a Banach space. A subtler generalization is to consider the trace of a family of operators as a kind of "twisted" dimension. This occurs significantly in representation theory, where the character of a representation is the trace of the representation, hence a scalar-valued function on a group whose value on the identity is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: The other values of the character can be viewed as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example