Editing Ring homomorphism (section)

== 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 {{nowrap|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 its 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 [[division ring]] 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.