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In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape".
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems.
The homomorphism, h, is a group monomorphism; i.e., h is injective ... The commutativity of H is needed to prove that h + k is again a group homomorphism.
A function: between two topological spaces is a homeomorphism if it has the following properties: . is a bijection (one-to-one and onto),; is continuous,; the inverse function is continuous (is an open mapping).
If : is a homomorphism of groups, the universal property of the cokernel is satisfied by the natural map / (), where () is the normalization of the image of . The snake lemma fails with this definition of cokernel: The connecting homomorphism can still be defined, and one can write down a sequence as in the statement of the snake lemma.
Given a ring homomorphism R → S of commutative rings and an S-module M, an R-linear map θ: S → M is called a derivation if for any f, g in S, θ(f g) = f θ(g) + θ(f) g. If S, T are unital associative algebras over a ring R, then an algebra homomorphism from S to T is a ring homomorphism that is also an R-module homomorphism.
A ring endomorphism is a ring homomorphism from a ring to itself. A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function on the underlying sets.
A map : is a local homomorphism if and only if : is a local homeomorphism and () is an open subset of . Every fiber of a local homeomorphism f : X → Y {\displaystyle f:X\to Y} is a discrete subspace of its domain X . {\displaystyle X.}