Definition 1: Bijective
A bijective map is a map f from set X to a set Y with the property that for every y in Y, there is exactly one x in X such that f(x) = y.
Definition 2: Homomorphism
A homomorphism is a map from one algebraic structure to another of the same type that preserves certain properties.
Definition 3: Isomorphism
An isomorphism is a bijective map f such that both f and its inverse f-1 are homomorphisms.
Definition 4: Automorphism
An automorphism is an isomorphism from a mathematical object to itself.
Definition 5: Ring Homomorphism
A ring homomorphism is a mapping between two rings which preserves the operations of addition and multiplication.
If R,S are rings and f: is the mapping R → S, the the following properties hold:
(1) f(a+b) = f(a) + f(b) for all a,b ∈ R
(2) f(ab) = f(a)f(b) for all a,b ∈ R
(3) f(1) = 1
Definition 6: Field Automorphism
A field automorphism is a bijective ring homomorphism from a field to itself.