Commutative Algebra/Primary decomposition or Lasker–Noether theory

The following theory was originally developed by world chess champion Emmanuel Lasker in his doctoral thesis under David Hilbert and then greatly simplified (and generalised to noetherian rings) by Emmy Noether.
Primary ideals.
Clearly, every prime ideal is primary.
We have the following characterisations:
Theorem 19.5 (characterisations of primary ideals):
Let formula_1. The following are equivalent:
Proof 1:
1. formula_8 2.: Let formula_2 be primary. Assume formula_3 and neither formula_4 nor formula_5. Since formula_13, formula_14 for a suitable formula_15. Since formula_16 and formula_17, formula_18 for a suitable formula_19.
2. formula_8 3.: Let formula_21 be a zerodivisor of formula_7, that is, formula_23 for a certain formula_24 such that formula_25. Hence formula_26, that is, formula_27 for a suitable formula_28.
3. formula_8 1.: Let formula_3. Then either formula_4 or formula_5 or formula_33 is a zerodivisor within formula_7, which is why formula_35 for a suitable formula_28.
Proof 2:
1. formula_8 3.: Let formula_2 be primary, and let formula_33 be a zerodivisor within formula_7. Then formula_3 for a formula_17 and hence formula_43 for a suitable formula_28.
3. formula_8 2.: Let formula_3. Assume neither formula_4 nor formula_5. Then both formula_33 and formula_50 are zerodivisors in formula_7, and hence are nilpotent, which is why formula_52 for suitable formula_53 and hence formula_54.
2. formula_8 1.: Let formula_3. Assume not formula_4 and not formula_5. Then in particular formula_59, that is, formula_14 for suitable formula_28.
Theorem 19.6:
If formula_1 is any primary ideal, then formula_63 is prime.
Proof:
Let formula_64. Then formula_65 for a suitable formula_66. Hence either formula_67 and thus formula_68 or formula_69 for a suitable formula_70 and hence formula_59.
Existence.
Existence in the Noetherian case.
Following the exposition of Zariski, Samuel and Cohen, we deduce the classical Noetherian existence theorem from two lemmas and a definition.
Definition 19.7:
An ideal formula_72 is called irreducible if and only if it can not be written as the intersection of finitely many proper superideals.
Lemma 19.8:
In a Noetherian ring, every irreducible ideal is primary.
Proof:
Assume there exists an irreducible ideal formula_73 which is not primary. Since formula_73 is not primary, there exist formula_75 such that formula_76, but neither formula_77 nor formula_78 for any formula_66. We form the ascending chain of ideals
this chain is ascending because formula_81. Since we are in a Noetherian ring, this chain eventually stabilizes at some formula_70; that is, for formula_83 we have formula_84. We now claim that
Indeed, formula_86 is obvious, and for formula_87 we note that if formula_88, then
for suitable formula_90 and formula_91, which is why formula_92, hence formula_93, since formula_76 thus formula_95, formula_96, hence formula_97 and formula_98. Therefore formula_99.
Furthermore, by the choice of formula_100 and formula_101 both formula_102 and formula_103 are proper superideals, contradicting the irreducibility of formula_73.
Lemma 19.9:
In a Noetherian ring, every ideal can be written as the finite intersection of irreducible ideals.
Proof:
Assume otherwise. Consider the set of all ideals that are not the finite intersection of irreducible ideals. If we are given an ascending chain within that set
this chain has an upper bound, since it stabilizes as we are in a Noetherian ring. We may hence choose a maximal element formula_73 among all ideals that are not the finite intersection of irreducible ideals. formula_73 itself is thus not irreducible. Hence, it can be written as the intersection of strict superideals; that is
for appropriate formula_109. Since formula_73 is maximal, each formula_111 is a finite intersection of irreducible ideals, and hence so is formula_73, which contradicts the choice of formula_73.
Proof:
Combine lemmas 19.8 and 19.9.
Minimal decomposition.
In fact, once we have a primary composition for a given ideal, we can find a minimal primary decomposition of that ideal. But before we prove that, we need a general fact about radicals first.
Lemma 19.12:
Let formula_114 be ideals. Then
One could phrase this lemma as "radical interchanges with finite intersections".
Proof:
formula_8:
formula_118:
Let formula_119. For each formula_120, choose formula_121 such that formula_122. Set
Then formula_124, hence formula_125.
 Note that for infinite intersections, the lemma need "not" (!!!) be true.
Proof 1:
First of all, we may exclude all primary ideals formula_126 for which
the intersection won't change if we do that, for intersecting with a superset changes nothing in general.
Then assume we are given a decomposition
and for a fixed prime ideal formula_129 set
due to theorem 19.6,
We claim that formula_132 is primary, and formula_133. For the first claim, note that by the previous lemma
For the second claim, let formula_135. If formula_136 there is nothing to prove. Otherwise let formula_137. Then there exists formula_138 such that formula_139, and hence formula_140 for a suitable formula_28. Thus formula_142, and hence formula_143 for all formula_120 and suitable formula_121. Pick
Then formula_147. Hence, formula_132 is primary.
Uniqueness properties.
In general, we don't have uniqueness for primary decompositions, but still, any two primary decompositions of the same ideal in a ring look somewhat similar. The classical first and second uniqueness theorems uncover some of these similarities.
Proof:
We begin by deducing an equation. According to theorem 19.2 and lemma 19.12,
Now we fix formula_150 and distinguish a few cases.
In conclusion, we find
Assume first that formula_164 is prime. Then the prime avoidance lemma implies that formula_164 is contained within one of the formula_166, formula_160, and since formula_168, formula_169.
Let now formula_166 for formula_171 be given. Since the given primary decomposition is minimal, we find formula_172 such that formula_153, but formula_174. In this case, formula_169 by the above equation.
This theorem motivates and enables the following definition:
We now prove two lemmas, each of which will below yield a proof of the second uniqueness theorem (see below).
Lemma 19.16:
Let formula_72 be an ideal which has a primary decomposition
and let again formula_178 for all formula_120. If we define
then formula_181 is an ideal of formula_182 and formula_183.
Proof:
Let formula_184. There exists formula_185 such that formula_186 without formula_187, and a similar formula_188 with an analogous property in regard to formula_189. Hence formula_190, but not formula_191 since formula_166 is prime. Also, formula_193. Hence, we have an ideal.
Let formula_194. There exists formula_185 such that
In particular, formula_197. Since no power of formula_198 is in formula_199, formula_200.
Lemma 19.17:
Let formula_201 be multiplicatively closed, and let
be the canonical morphism. Let formula_73 be a decomposable ideal, that is
for primary formula_150, and number the formula_150 such that the first formula_207 formula_150 have empty intersection with formula_209, and the others nonempty intersection. Then
Proof:
We have
by theorem 9.?. If now formula_212, lemma 9.? yields formula_213. Hence,
Application of formula_215 on both sides yields
and
since formula_87 holds for general maps, and formula_219 means formula_220, where formula_221 and formula_222; thus formula_223, that is formula_224. This means that
Hence formula_226, and since no power of formula_227 is in formula_150 (formula_209 is multiplicatively closed and formula_230), formula_151.
Note that applied to reduced sets consisting of only one prime ideal, this means that if all prime subideals of a prime ideal formula_232 belonging to formula_73 also belong to formula_73, then the corresponding formula_199 is predetermined.
Proof 1 (using lemma 19.16):
We first reduce the theorem down to the case where formula_236 is the set of all prime subideals belonging to formula_73 of a prime ideal that belongs to formula_73. Let formula_236 be any reduced system. For each maximal element of that set formula_240 (w.r.t. inclusion) define formula_241 to be the set of all ideals in formula_236 contained in formula_240. Since formula_236 is finite,
this need not be a disjoint union (note that these are not maximal "ideals"!). Hence
Hence, let formula_129 be an ideal belonging to formula_73 and let formula_249 be an isolated system of subideals of formula_129. Let formula_251 be all the primary ideals belonging to formula_73 not in formula_236. For those ideals, we have formula_254, and hence we find formula_255. For each formula_256 take formula_257 large enough so that formula_258. Then
which is why
formula_260.
From this follows that
where formula_2 is the element in the primary decomposition of formula_73 to which formula_129 is associated, since clearly for each element formula_100 of the left hand side, formula_266 and thus formula_267, but also formula_268. But on the other hand, formula_269 implies formula_270. Hence for any such formula_271 lemma 19.16 implies
which in turn implies
Proof 2 (using lemma 19.17):
Let formula_274 be an isolated system of prime ideals belonging to formula_73. Pick
which is multiplicatively closed since it's the intersection of multiplicatively closed subsets. The primary ideals of the decomposition of formula_73 which correspond to the formula_278 are precisely those having empty intersection with formula_209, since any other primary ideal formula_150 in the decomposition of formula_73 must contain an element outside all formula_274, since otherwise its radical would be one of them by isolatedness. Hence, lemma 19.17 gives
and we have independence of the particular decomposition.
Characterisation of prime ideals belonging to an ideal.
The following are useful further theorems on primary decomposition.
First of all, we give a proposition on general prime ideals.
Proof:
Since the product is contained in the intersection, it suffices to prove the theorem under the assumption that formula_284.
Indeed, assume none of the formula_114 is contained in formula_129. Choose formula_287 for formula_288. Since formula_129 is prime, formula_290. But it's in the product, contradiction.
This proposition has far-reaching consequences for primary decomposition, given in Corollary 19.22. But first, we need a lemma.
Lemma 19.21:
Let formula_1 be a primary ideal, and assume formula_292 is prime such that formula_293. Then formula_294.
Proof:
If formula_67, then formula_296.
Proof:
The first assertion follows from proposition 19.20 and lemma 19.21. The second assertion follows since any prime ideal belonging to formula_73 contains formula_73.