Let be a ring homomorphism. Then, directly from these definitions, one can deduce: •
f(0
R) = 0
S. •
f(−
a) = −
f(
a) for all
a in
R. • For any
unit a in
R,
f(
a) is a unit element such that . 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 of
f, denoted im(
f), is a subring of
S. • The
kernel of
f, defined as , 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 of
S divides the characteristic of
R. This can sometimes be used to show that between certain rings
R and
S, no ring homomorphism exists. • If
Rp is the smallest
subring contained in
R and
Sp is the smallest subring contained in
S, then every ring homomorphism induces a ring homomorphism . • If
R is a
division ring and
S is not the
zero ring, then is injective. • If both
R and
S are
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 −1(
I) is an ideal of
R. • If
R and
S are commutative and
P is a
prime ideal of
S then −1(
P) is a prime ideal of
R. • If
R and
S are commutative,
M is a
maximal ideal of
S, and is surjective, then −1(
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 is surjective, then ker(
f) is a
maximal ideal of
R. • If is surjective,
P is prime (maximal) ideal in
R and , then
f(
P) is prime (maximal) ideal in
S. Moreover, • The composition of ring homomorphisms and is a ring homomorphism . • For each ring
R, the identity map is a ring homomorphism. • Therefore, the class of all rings together with ring homomorphisms forms a category, the
category of rings. • The zero map 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 . This says that the ring of integers is an
initial object in the
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 ==