By Ivan Fesenko

Commutative algebra

This path is an creation to modules over jewelry, Noetherian modules, distinctive factorization domain names and polynomial earrings over them, modules over significant excellent domain names; spectrum of earrings and their interpretations, localization, extensions of fields and creation to Galois conception.

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**Example text**

2. Let A = Z and I = aZ for a > 1. Then lim A/I n is isomorphic to the ring ←− of a -adic numbers Za = n 0 cn an , 0 cn < a . In this example, φ is injective, but not surjective. In particular, if a is prime p, the corresponding ring Zp is one of central objects in number theory. The product of all Zp where p ranges through all positive primes, is called the profinite completion of Z and is denoted Z. 3. More generally, we can similarly define the completion lim A/In of a ring A ←− with respect to a decreasing sequence of its ideals In .

If S = A \ P for a prime ideal P of A, then we write MP = MS and call it the localization of M with respect to P . Lemma. AS ⊗A M MS , f : (a/s, m) → am/s. Proof. The map is A -bilinear, so we obtain AS ⊗A M → MS . Define a map MS → AS ⊗A M , m/s → 1/s ⊗ m. This is well defined: if sm = s m then 1/s ⊗ m = s /(ss ) ⊗ m = 1/(ss )s ⊗ s m = 1/(ss ) ⊗ sm = 1/s ⊗ m . This map is the inverse to f . 10. 1. Definition. Let I be an ideal of a ring A. Consider the set of sequences (an ), n 0, of elements of A , such that an − am ∈ I m for all n m 0.

Thus, (M/N )S MS /NS . 10. If S = A \ P for a prime ideal P of A, then we write MP = MS and call it the localization of M with respect to P . Lemma. AS ⊗A M MS , f : (a/s, m) → am/s. Proof. The map is A -bilinear, so we obtain AS ⊗A M → MS . Define a map MS → AS ⊗A M , m/s → 1/s ⊗ m. This is well defined: if sm = s m then 1/s ⊗ m = s /(ss ) ⊗ m = 1/(ss )s ⊗ s m = 1/(ss ) ⊗ sm = 1/s ⊗ m . This map is the inverse to f . 10. 1. Definition. Let I be an ideal of a ring A. Consider the set of sequences (an ), n 0, of elements of A , such that an − am ∈ I m for all n m 0.