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is called an
ideal
of R.
An ideal is a subring. Note that if the ring R is commutative, then we can replace
condition (2) in the definition of an ideal by
(2¢) ra Œ I for all r Œ R and a Œ I.
Definition.
Let R and R¢ be rings. A map h : R Æ R¢ is said to be a (ring)
homomor-
phism
if
(
)
=
()
+
()
h a
+
b
h a
h b
and
()
=
() (
,
hab
hahb
for all a, b ΠR. If h is a bijection, then h is called a (ring)
isomorphism
.
Definition.
Let h : R Æ R¢ be a homomorphism between rings. The
kernel
of h, ker
h, and the
image
of h, im(h), are defined by
{
()
=
0
ker h
=Œ
r
R
h r
and
()
=
{
()
}
.
im h
h r
r
Œ
R
B.6.1. Theorem.
(1) The image of a ring homomorphism is a subring.
(2) The kernel of a homomorphism is an ideal.
(3) A ring homomorphism is an isomorphism if and only if its kernel is
0
.
Proof.
See [Fral67].
Definition.
Let I be an ideal in a ring (R,+,·). The
factor
or
quotient ring
of the ring
R by the ideal I, denoted by (R/I,+,·), is defined as follows: The additive group (R/I,+)
is just the quotient group of (R,+) by the subgroup (I,+). The operation · is defined by
(
)
◊
(
)
=+.
r
+
I
s
+
I
rs
I
Using the definition of an ideal it is easy to check that the quotient ring of a ring
R by an ideal I is in fact a ring and that the map
RRI
Æ
that sends an element into its coset is a surjective ring homomorphism with kernel I.
In analogy with the integers one can define a notion of congruence.
Definition.
Let I be an ideal in a commutative ring R. Let a, b ΠR. We say that a
and b are
congruent modulo I
, or
mod I
, and write
