Editing Ring (mathematics) (section)

=== Homomorphism ===
{{main|Ring homomorphism}}
A '''[[ring homomorphism|homomorphism]]''' from a ring {{math|(''R'', +, '''⋅''')}} to a ring {{math|(''S'', ‡, ∗)}} is a function {{mvar|f}} from {{mvar|R}} to&nbsp;{{mvar|S}} that preserves the ring operations; namely, such that, for all {{math|''a''}}, {{math|''b''}} in {{mvar|R}} the following identities hold:
:<math>\begin{align}
& f(a+b) = f(a) \ddagger f(b) \\
& f(a\cdot b) = f(a)*f(b) \\
& f(1_R) = 1_S
\end{align}</math>

If one is working with {{nat|rngs}}, then the third condition is dropped.

A ring homomorphism {{mvar|f}} is said to be an '''[[isomorphism]]''' if there exists an inverse homomorphism to {{mvar|f}} (that is, a ring homomorphism that is an [[inverse function]]), or equivalently if it is [[bijection|bijective]].

Examples:
* The function that maps each integer {{mvar|x}} to its remainder modulo {{math|4}} (a number in {{math|{{mset|0, 1, 2, 3}}}}) is a homomorphism from the ring {{tmath|\Z}} to the quotient ring {{tmath|\Z/4\Z}} ("quotient ring" is defined below).
* If {{mvar|u}} is a unit element in a ring {{mvar|R}}, then <math>R \to R, x \mapsto uxu^{-1}</math> is a ring homomorphism, called an [[inner automorphism]] of {{mvar|R}}.
* Let {{mvar|R}} be a commutative ring of prime characteristic {{mvar|p}}. Then {{math|''x'' ↦ {{itco|''x''}}{{sup|''p''}}}} is a ring endomorphism of {{mvar|R}} called the [[Frobenius homomorphism]].
* The [[Galois group]] of a field extension {{math|''L'' / ''K''}} is the set of all automorphisms of {{mvar|L}} whose restrictions to {{mvar|K}} are the identity.
* For any ring {{mvar|R}}, there are a unique ring homomorphism {{tmath|\Z \mapsto R}} and a unique ring homomorphism {{math|''R'' → 0}}.
* An [[epimorphism]] (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map {{tmath|\Z\to\Q}} is an epimorphism.
* An algebra homomorphism from a {{mvar|k}}-algebra to the [[endomorphism algebra]] of a vector space over {{mvar|k}} is called a [[algebra representation|representation of the algebra]].

Given a ring homomorphism {{math|''f'' : ''R'' → ''S''}}, the set of all elements mapped to 0 by {{mvar|f}} is called the [[kernel of a ring homomorphism|kernel]] of&nbsp;{{mvar|f}}. The kernel is a two-sided ideal of&nbsp;{{mvar|R}}. The image of&nbsp;{{mvar|f}}, on the other hand, is not always an ideal, but it is always a subring of&nbsp;{{mvar|S}}.

To give a ring homomorphism from a commutative ring {{mvar|R}} to a ring {{mvar|A}} with image contained in the center of {{mvar|A}} is the same as to give a structure of an [[associative algebra|algebra]] over {{mvar|R}} to&nbsp;{{mvar|A}} (which in particular gives a structure of an {{mvar|A}}-module).