algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, the coimage of a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

f : A \rightarrow B

is the quotient :

\text f = A/\ker(f)

of the domain by the kernel. The coimage is canonically isomorphic to the

image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

by the first isomorphism theorem, when that theorem applies. More generally, in

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, the coimage of a

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

is the dual notion of the image of a morphism. If

f : X \rightarrow Y

, then a coimage of

f

(if it exists) is an epimorphism

c : X \rightarrow C

such that #there is a map

f_c : C \rightarrow Y

with

f =f_c \circ c

, #for any epimorphism

z : X \rightarrow Z

for which there is a map

f_z : Z \rightarrow Y

with

f =f_z \circ z

, there is a unique map

h : Z \rightarrow C

such that both

c =h \circ z

and

f_z =f_c \circ h

References

* Abstract algebra Isomorphism theorems Category theory pl:Twierdzenie o izomorfizmie#Pierwsze twierdzenie {{Linear-algebra-stub

See also

References