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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 in which 1 = 0.) The ideals we are interested in are th
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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, any
nonempty collection of ideals has a maximal elemen
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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.).
Denition 6.1. A collection of subsets of a set S satises
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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,
K = B B 1
where B 1 denotes the set of all inverses o
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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
(2) bn is integral over a for some n 1
(3) bn aB for so
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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 is a eld and a = 0 A. Then B contains the
inverse a1 of
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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. Suppose that A is a subring of B . We say that B is
an i
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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 A
is Noetherian. So, the denition of an associated pri
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MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA
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
=
particular:
(M A N )p Mp Ap Np
=
3.5. local properties. A proper
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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. They dont
assume 0 S (as in my Def 1.19). So, we will
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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 have
a concrete description so that we can distinguish be
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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 )
Here E (M ) is the ring of all endomorphisms of the addi
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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 1.29. For any ring A let Sp ec(A) denote the set of prime
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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 b p
Prop osition 1.13. An ideal a is prime i A/a is an