Colinear map
Encyclopedia
In coalgebra
theory, the notion of colinear map is dual to the notion for linear map of vector space
, or more generally, for morphism between R-module. Specifically, let R be a ring
, M,N,C be R-modules, and
be right C-comodule
s. Then an R-linear map
is called a (right) comodule morphism, or (right) C-colinear, if
Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
theory, the notion of colinear map is dual to the notion for linear map of vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, or more generally, for morphism between R-module. Specifically, let R be a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, M,N,C be R-modules, and
be right C-comodule
Comodule
In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.-Formal definition:...
s. Then an R-linear map
is called a (right) comodule morphism, or (right) C-colinear, if