Editing Ring homomorphism (section)

== Examples ==
* The function {{nowrap|''f'' : '''Z''' → '''Z'''/''n'''''Z'''}}, defined by {{nowrap|1=''f''(''a'') = [''a'']<sub>''n''</sub> = ''a'' mod ''n''}} is a [[surjective]] ring homomorphism with kernel ''n'''''Z''' (see ''[[Modular arithmetic]]'').
* The [[complex conjugation]] {{nowrap|'''C''' → '''C'''}} is a ring homomorphism (this is an example of a ring automorphism).
* For a ring ''R'' of prime characteristic ''p'', {{nowrap|''R'' → ''R'', ''x'' → {{itco|''x''}}<sup>''p''</sup>}} is a ring endomorphism called the [[Frobenius endomorphism]].
* If ''R'' and ''S'' are rings, the zero function from ''R'' to ''S'' is a ring homomorphism if and only if ''S'' is the [[zero ring]]  (otherwise it fails to map 1<sub>''R''</sub> to 1<sub>''S''</sub>).  On the other hand, the zero function is always a {{not a typo|rng}} homomorphism.
* If '''R'''[''X''] denotes the ring of all [[polynomial]]s in the variable ''X'' with coefficients in the [[real number]]s '''R''', and '''C''' denotes the [[complex number]]s, then the function {{nowrap|''f'' : '''R'''[''X''] → '''C'''}} defined by {{nowrap|1=''f''(''p'') = ''p''(''i'')}} (substitute the imaginary unit ''i'' for the variable ''X'' in the polynomial ''p'') is a surjective ring homomorphism. The kernel of ''f'' consists of all polynomials in '''R'''[''X''] that are divisible by {{nowrap|{{itco|''X''}}<sup>2</sup> + 1}}.
* If {{nowrap|''f'' : ''R'' → ''S''}} is a ring homomorphism between the rings ''R'' and ''S'', then ''f'' induces a ring homomorphism between the [[matrix ring]]s {{nowrap|M<sub>''n''</sub>(''R'') → M<sub>''n''</sub>(''S'')}}.
* Let ''V'' be a vector space over a field ''k''. Then the map {{nowrap|''ρ'' : ''k'' → End(''V'')}} given by {{nowrap|1=''ρ''(''a'')''v'' = ''av''}} is a ring homomorphism. More generally, given an abelian group ''M'', a module structure on ''M'' over a ring ''R'' is equivalent to giving a ring homomorphism {{nowrap|''R'' → End(''M'')}}.
* A unital [[algebra homomorphism]] between unital [[associative algebra]]s over a commutative ring ''R'' is a ring homomorphism that is also [[module homomorphism|''R''-linear]].