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MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA
1. Ideals
In this course, all rings A will be commutative with unity 1. An ideal
a A is allowed to be all of A. So A/a = 0 is a ring. (Zero is the only
ring
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7. Noetherian rings
A ring is Noetherian if it satised the ascending chain condition for
ideal. Equivalently, every ideal is nitely generated. Equivalently,
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MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA
6. Chain conditions
I will skip the details of this section and explain them as needed in
the next two sections (on Noetherian and Artinian rings resp.).
De
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5.4. valuation rings. Suppose that K is any eld. Then a valuation
ring of K is any subring B so that for any nonzero x K either x B
or x1 B . In other words
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Lemma 5.20. Suppose a is an ideal in A and b B where B is an
integral extension of A. Then the following are equivalent.
(1) b is integral over the ideal a
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5.2. going up.
Proposition 5.11. Suppose that A, B are integral domains and B is
an integral extension of A. Then B is a eld i A is a eld.
Proof. Suppose B
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5. Integral dependence and valuation
This section is about integral extensions and the integral closure of
a ring.
5.1. integral extensions.
Denition 5.1. S
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4. Primary decomposition
This chapter of Atiyah-MacDonald is dierent from other treatments
of primary decomposition because it does not assume that the ring
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3.4. review.
x
= 0 in S 1 M xu = 0 for some u S
s
and (Corollary 3.9):
S 1 M S 1 A M
=
Lemma 3.12 (A-M 3.7). S 1 (M A N ) S 1 M S 1 A S 1 N . In
=
particula
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3. Fractions in rings and modules
Atiyah and MacDonald dene multiplicative sets to be subsets S A
of a ring so that 1 S and S is closed under multiplication
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2.5. tensor product.
2.5.1. denition and basic properties. The best description of the tensor product is given by the universal property. But, we need to ha
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2. Modules
2.1. basic concepts. One way to dene a module over a ring A is to
say that it is an additive group M together with a ring homomorphism
A E (M )
H
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1.3. Basic Algebraic Geometry. I will use exercises 15,17,21 from
the book and other example from other books to review basic algebraic
geometry.
Denition 1.
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MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA
1.2. prime ideals.
Denition 1.12. A prime ideal is a proper ideal whose complement
is closed under multiplication.
This is equivalent to saying:
ab p a p or