Graphics Reference
In-Depth Information
B.5.10. Lemma.
(hom (G,H),+) is an abelian group.
Proof.
Straightforward.
B.5.11. Lemma.
Let G be any group that is isomorphic to
Z
. If h Πhom (G,G), then
there is a unique integer k such that h(g) = kg for all g in G.
Proof.
Since G is isomorphic to
Z
, there is some g
0
in G so that G =
Z
g
0
. It fol-
lows that h(g
0
) = kg
0
for some integer k, because {g
0
} is a basis for G. The fact that
o(g
0
) =•implies that the integer k is unique. Now let g be any element of G. Again,
there is some integer t with g = tg
0
. Thus,
()
=
(
)
=
()
=
(
)
=
(
)
=
hg
htg
thg
tkg
ktg
kg
,
0
0
0
0
and the lemma is proved.
Lemma B.5.11 implies that if G ª
Z
, then hom (G,G) =
Z
1
G
ª
Z
.
B.6
Rings
Definition.
A
ring
is a triple (R,+,·) where R is a set and + and · are two binary oper-
ations on R, called addition and multiplication, respectively, satisfying the following:
(1) (R,+) is an abelian group.
(2) The multiplication · is associative.
(3) For all a, b, c ΠR we have
(a) (
left distributativity
)
(b) (
right distributativity
)
(
)
=◊
(
)
+◊
(
)
abc
◊
+
ab ac
(
)
◊= ◊
(
)
+◊
(
)
abc ac bc
+
Two standard examples of rings are
Z
and
Z
n.
Definition.
A ring in which the multiplication is commutative is called a
commuta-
tive ring
. A ring with a multiplicative identity is called a
ring with unity
. An element
of a ring with unity is called a
unit
if it has a multiplicative inverse in R.
Definition.
Let (R,+,·) be a ring. If A is a subset of R and if (A,+,·) is a ring, then
(A,+,·) is called a
subring
of R.
Note.
From now on, like in the case of groups, we shall not explicitly mention the
operations + and · for a ring(R,+,·) and simply refer to “the ring R.” Products “r · s”
will be abbreviated to “rs.”
Definition.
A subset I of a ring R with the property that
(1) I is an additive subgroup of R (equivalently, a - b ΠI for all a, b ΠI), and
(2) ra, ar ΠI for all r ΠR and a ΠI,
