1.11. RINGS AND FIELDS
77
Example 1.11.5.
(a)
Z
is a subring of
Q
.
(b)
The ring of 3by3 matrices with
rational
entries is a subring of
the ring of 3by3 matrices with
real
entries.
(c)
The set of
upper triangular
3by3 matrices with real entries is a
subring of the ring of
all
3by3 matrices with real entries.
(d)
The set of
continuous
functions from
R
to
R
is a subring of the
ring of
all
functions from
R
to
R
.
(e)
The set of
rational–valued
functions on a set
X
is a subring of
the ring of
real–valued
functions on
X
.
A second way in which two rings may be related to one another is
by a
homomorphism
. A map
f
W
R
!
S
between two rings is said to
be a
homomorphism
if it “respects” the ring structures. More explicitly,
f
must take sums to sums and products to products. In notation,
f .a
C
b/
D
f .a/
C
f .b/
and
f .ab/
D
f .a/f .b/
for all
a; b
2
R
. A bijective
homomorphism is called an
isomorphism
.
Example 1.11.6.
(a)
The map
f
W
Z
!
Z
n
defined by
f .a/
D
OEaŁ
is a homomor
phism of rings, because
f .a
C
b/
D
OEa
C
bŁ
D
OEaŁ
C
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 Fall '08
 EVERAGE
 Algebra, Matrices, Homomorphism, Epimorphism, real entries

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