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Proof.
The theorem follows from Corollary 10.8.6 and Theorems 10.8.15 and 10.8.16.
Finally, to press home the close relationship between algebra and geometry further,
we show how some other algebraic operations on ideals correspond to set-theoretic
operations on varieties and then summarize everything in Table 10.8.1. We first need
one more definition to capture the algebraic analog of the difference of two varieties.
Definition.
Let I and J be ideals in k[X
1
,X
2
,...,X
n
]. Define the
ideal quotient
or
colon
ideal
, I : J, by
=Œ
[
]
{
}
I J
:
p
k X
,
X
,...,
X
pq
Œ
I for all q
Œ
J
.
12
n
10.8.18. Proposition.
The ideal quotient of two ideals I and J in k[X
1
,X
2
,...,X
n
] is
an ideal that contains I.
Proof.
Straightforward.
10.8.18. Theorem.
Let I and J be ideals in k[X
1
,X
2
,...,X
n
].
Table 10.8.1
The correspondence between affine alge-
braic and geometric concepts.
Algebra Geometry
radical ideals I, J
Õ
k[X
1
,X
2
,..., X
n
] varieties V, W
Õ
k
n
(the field k is assumed to be algebraically closed)
Radical ideal
Varieties
I
V(I)
I(
V
)
V
Inclusion of ideals
Reverse inclusion of
varieties
I
Õ
J
VI
()
VJ
()
Addition of ideals
Intersection of varieties
I
+
J
V(I)
«
V(J)
V
«
W
()
+
()
I
VW
Product of ideals
Union of varieties
I J
V(I)
»
V(J)
V
»
W
I
()( )
VW
Intersection of ideals
Union of varieties
I
«
J
V(I)
»
V(J)
I(
V
)
«
I(
W
)
V
»
W
Quotient of ideals
Differ
ence of va
rieties
I : J
V
(I)
-
V(J
)
I(
V
):I(
W
)
V
-
W
Prime ideal
Irreducible variety
Maximal ideal
Point of affine space
