algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, the coimage of a homomorphism :

f : A \rightarrow B

is the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

\text f = A/\ker(f)

of the domain by the

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...

. The coimage is canonically isomorphic to the

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

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 that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, the coimage of a

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...

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

*{{Mitchell TOC Abstract algebra Isomorphism theorems Category theory pl:Twierdzenie o izomorfizmie#Pierwsze twierdzenie

See also

References