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{{Short description|Structure-preserving function between two rings}}
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In [[mathematics]], a '''ring homomorphism''' is a structure-preserving [[function (mathematics)|function]] between two [[ring (mathematics)|rings]]. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function <math>f:R \to S</math> that preserves addition, multiplication and [[multiplicative identity]]; that is,{{sfn|ps=|Artin|1991|p=353}}{{sfn|ps=|Eisenbud|1995|p=12}}{{sfn|ps=|Jacobson|1985|p=103}}{{sfn|ps=|Lang|2002|p=88}}{{sfn|ps=|Hazewinkel|2004|p=3}}
:<math>\begin{align}
f(a+b)&= f(a) + f(b),\\
f(ab) &= f(a)f(b), \\
f(1_R) &= 1_S,
\end{align}</math>
for all <math>a,b</math> in <math>R.</math>

These conditions imply that additive inverses and the additive identity are preserved too.

If in addition {{itco|''f''}} is a [[bijection]], then its [[Inverse function|inverse]] {{itco|''f''}}<sup>−1</sup> is also a ring homomorphism.  In this case, {{itco|''f''}} is called a '''ring isomorphism''', and the rings ''R'' and ''S'' are called ''isomorphic''. From the standpoint of ring theory, isomorphic rings have exactly the same properties.

If ''R'' and ''S'' are <!--NOT A MISSPELLING-->[[rng (algebra)|rng]]s<!--NOT A MISSPELLING-->, then the corresponding notion is that of a '''<!--NOT A MISSPELLING-->rng<!--NOT A MISSPELLING--> homomorphism''',{{efn|Some authors use the term "ring" to refer to structures that do not require a multiplicative identity; instead of <!--NOT A MISSPELLING-->"rng"<!--NOT A MISSPELLING-->, "ring", and <!--NOT A MISSPELLING-->"rng<!--NOT A MISSPELLING--> homomorphism", they use the terms "ring", "ring with identity", and "ring homomorphism", respectively.  Because of this, some other authors, to avoid ambiguity, explicitly specify that rings are unital and that homomorphisms preserve the identity.}} defined as above except without the third condition ''f''(1<sub>''R''</sub>) = 1<sub>''S''</sub>.  A <!--NOT A MISSPELLING-->rng<!--NOT A MISSPELLING--> homomorphism between (unital) rings need not be a ring homomorphism.

The [[function composition|composition]] of two ring homomorphisms is a ring homomorphism. It follows that the rings forms a [[category (mathematics)|category]] with ring homomorphisms as [[morphism]]s (see [[Category of rings]]).
In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.

== Properties ==
Let {{nowrap|''f'' : ''R'' → ''S''}} be a ring homomorphism. Then, directly from these definitions, one can deduce:
* ''f''(0<sub>''R''</sub>) = 0<sub>''S''</sub>.
* ''f''(−''a'') = −''f''(''a'') for all ''a'' in ''R''.
* For any [[unit (ring theory)|unit]] ''a'' in ''R'', ''f''(''a'') is a unit element such that {{math|1=''f''(''a'')<sup>−1</sup> = ''f''(''a''<sup>−1</sup>)}} . In particular, ''f'' induces a [[group homomorphism]] from the (multiplicative) group of units of ''R'' to the (multiplicative) group of units of ''S'' (or of im(''f'')).
* The [[Image (mathematics)|image]] of ''f'', denoted im(''f''), is a subring of ''S''.
* The [[kernel (algebra)|kernel]] of ''f'', defined as {{nowrap|1=ker(''f'') = {{mset|''a'' in ''R'' {{pipe}} ''f''(''a'') {{=}} 0<sub>''S''</sub>}}}}, is a [[two-sided ideal]] in ''R''. Every two-sided ideal in a ring ''R'' is the kernel of some ring homomorphism.
* A homomorphism is injective if and only if kernel is the [[zero ideal]].
* The [[characteristic (algebra)|characteristic]] of ''S'' [[divides]] the characteristic of ''R''.  This can sometimes be used to show that between certain rings ''R'' and ''S'', no ring homomorphism {{nowrap|''R'' → ''S''}} exists.
* If ''R<sub>p</sub>'' is the smallest [[subring]] contained in ''R'' and ''S<sub>p</sub>'' is the smallest subring contained in ''S'', then every ring homomorphism {{nowrap|''f'' : ''R'' → ''S''}} induces a ring homomorphism {{nowrap|''f<sub>p</sub>'' : ''R<sub>p</sub>'' → ''S<sub>p</sub>''}}.
* If ''R'' is a [[Field (mathematics)|field]] (or more generally a [[skew-field]]) and ''S'' is not the [[zero ring]], then {{itco|''f''}} is injective.
* If both ''R'' and ''S'' are [[Field (mathematics)|fields]], then im(''f'') is a subfield of ''S'', so ''S'' can be viewed as a [[field extension]] of ''R''.
* If ''I'' is an ideal of ''S'' then {{itco|''f''}}<sup>−1</sup>(''I'') is an ideal of ''R''.
* If ''R'' and ''S'' are commutative and ''P'' is a [[prime ideal]] of ''S'' then {{itco|''f''}}<sup>−1</sup>(''P'') is a prime ideal of ''R''.
* If ''R'' and ''S'' are commutative, ''M'' is a [[maximal ideal]] of ''S'', and {{itco|''f''}} is surjective, then {{itco|''f''}}<sup>−1</sup>(''M'') is a maximal ideal of ''R''.
* If ''R'' and ''S'' are commutative and ''S'' is an [[integral domain]], then ker(''f'') is a prime ideal of ''R''.
* If ''R'' and ''S'' are commutative, ''S'' is a field, and {{itco|''f''}} is surjective, then ker(''f'') is a [[maximal ideal]] of ''R''.
* If {{itco|''f''}} is surjective, ''P'' is prime (maximal) ideal in ''R'' and {{nowrap|1=ker(''f'') ⊆ ''P''}}, then ''f''(''P'') is prime (maximal) ideal in ''S''.

Moreover,
* The composition of ring homomorphisms {{nowrap|''S'' → ''T''}} and {{nowrap|''R'' → ''S''}} is a ring homomorphism {{nowrap|''R'' → ''T''}}.
* For each ring ''R'', the identity map {{nowrap|''R'' → ''R''}} is a ring homomorphism.
* Therefore, the class of all rings together with ring homomorphisms forms a category, the [[category of rings]].
* The zero map {{nowrap|''R'' → ''S''}} that sends every element of ''R'' to 0 is a ring homomorphism only if ''S'' is the [[zero ring]] (the ring whose only element is zero).
* For every ring ''R'', there is a unique ring homomorphism {{nowrap|'''Z''' → ''R''}}. This says that the ring of integers is an [[initial object]] in the [[Category (mathematics)|category]] of rings.
* For every ring ''R'', there is a unique ring homomorphism from ''R'' to the zero ring.  This says that the zero ring is a [[terminal object]] in the category of rings.
* As the initial object is not isomorphic to the terminal object, there is no [[zero object]] in the category of rings; in particular, the zero ring is not a zero object in the category of rings.

== 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 MISSPELLING-->rng<!--NOT A MISSPELLING--> 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]].

== Non-examples ==
* The function {{nowrap|''f'' : '''Z'''/6'''Z''' → '''Z'''/6'''Z'''}} defined by {{nowrap|1=''f''([''a'']<sub>6</sub>) = [4''a'']<sub>6</sub>}} is a <!--NOT A MISSPELLING-->ring<!--NOT A MISSPELLING--> homomorphism (and <!--NOT A MISSPELLING-->ring<!--NOT A MISSPELLING--> endomorphism), with kernel 3'''Z'''/6'''Z''' and image 2'''Z'''/6'''Z''' (which is isomorphic to '''Z'''/3'''Z''').
* There is no ring homomorphism {{nowrap|'''Z'''/''n'''''Z''' → '''Z'''}} for any {{nowrap|''n'' ≥ 1}}.
* If ''R'' and ''S'' are rings, the inclusion {{nowrap|''R'' → ''R'' × ''S''}} that sends each ''r'' to (''r'',0) is a rng homomorphism, but not a ring homomorphism (if ''S'' is not the zero ring), since it does not map the multiplicative identity 1 of ''R'' to the multiplicative identity (1,1) of {{nowrap|''R'' × ''S''}}.

== Category of rings ==
{{main|Category of rings}}

=== Endomorphisms, isomorphisms, and automorphisms ===
* 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.  If there exists a ring isomorphism between two rings ''R'' and ''S'', then ''R'' and ''S'' are called '''isomorphic'''.  Isomorphic rings differ only by a relabeling of elements.  Example: Up to isomorphism, there are four rings of order 4.  (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.)  On the other hand, up to isomorphism, there are eleven <!--NOT A MISSPELLING-->[[Rng (algebra)|rngs]]<!--NOT A MISSPELLING--> of order 4.
* A '''ring automorphism''' is a ring isomorphism from a ring to itself.

=== Monomorphisms and epimorphisms ===
Injective ring homomorphisms are identical to [[monomorphism]]s in the category of rings: If {{nowrap|''f'' : ''R'' → ''S''}} is a monomorphism that is not injective, then it sends some ''r''<sub>1</sub> and ''r''<sub>2</sub> to the same element of ''S''.  Consider the two maps ''g''<sub>1</sub> and ''g''<sub>2</sub> from '''Z'''[''x''] to ''R'' that map ''x'' to ''r''<sub>1</sub> and ''r''<sub>2</sub>, respectively; {{nowrap|''f'' ∘ ''g''<sub>1</sub>}} and {{nowrap|''f'' ∘ ''g''<sub>2</sub>}} are identical, but since {{itco|''f''}} is a monomorphism this is impossible.

However, surjective ring homomorphisms are vastly different from [[epimorphism]]s in the category of rings.  For example, the inclusion {{nowrap|'''Z''' ⊆ '''Q'''}} is a ring epimorphism, but not a surjection. However, they are exactly the same as the [[strong epimorphism]]s.

== See also ==
* [[Change of rings]]

== Notes ==
{{notelist}}

== Citations ==
{{reflist|2}}

== References ==
{{refbegin}}
* {{cite book
 | last=Artin | first=Michael | author-link=Michael Artin
 | year=1991
 | title=Algebra
 | location=Englewood Cliffs, N.J. | publisher=Prentice Hall
}}
* {{citation
 | last1=Atiyah | first1=Michael F. | author1-link=Michael Atiyah
 | last2=Macdonald | first2=Ian G.| author2-link=Ian G. Macdonald
 | year=1969
 | title=Introduction to commutative algebra
 | publisher=Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.
 | mr=0242802
}}
* {{cite book
 | last1=Bourbaki | first1=N. | author1-link=Nicolas Bourbaki
 | year=1998
 | title=Algebra I, Chapters 1–3
 | publisher=Springer
}}
* {{cite book
 | last=Eisenbud | first= David | author-link=David Eisenbud
 | year=1995
 | title=Commutative algebra with a view toward algebraic geometry
 | location=New York | publisher=[[Springer-Verlag]]
 | series=[[Graduate Texts in Mathematics]] | volume=150
 | mr=1322960 | isbn=0-387-94268-8
 | no-pp=true
 | page=xvi+785
}}
* {{cite book
 | last1=Hazewinkel | first1=Michiel | author-link1=Michiel Hazewinkel
 | year=2004
 | title=Algebras, rings and modules
 | publisher=[[Springer-Verlag]]
 | isbn=1-4020-2690-0
}}
* {{cite book
 | last1=Jacobson | first1=Nathan | author-link=Nathan Jacobson
 | year=1985
 | title=Basic algebra I
 | edition=2nd
 | isbn=9780486471891
 | url=https://store.doverpublications.com/0486471896.html
}}
* {{Lang Algebra|3rd}}
{{refend}}


[[Category:Ring theory|homomorphism]]
[[Category:Morphisms]]