Rings and modules by Ivan Fesenko

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.

Show description

Read Online or Download Rings and modules PDF

Similar group theory books

Representations of Groups: A Computational Approach

The illustration conception of finite teams has obvious quick progress lately with the advance of effective algorithms and computing device algebra structures. this can be the 1st booklet to supply an advent to the normal and modular illustration conception of finite teams with designated emphasis at the computational points of the topic.

Groups of Prime Power Order Volume 2 (De Gruyter Expositions in Mathematics)

This can be the second one of 3 volumes dedicated to hassle-free finite p-group conception. just like the 1st quantity, thousands of significant effects are analyzed and, in lots of instances, simplified. very important issues awarded during this monograph contain: (a) type of p-groups all of whose cyclic subgroups of composite orders are general, (b) category of 2-groups with precisely 3 involutions, (c) proofs of Ward's theorem on quaternion-free teams, (d) 2-groups with small centralizers of an involution, (e) class of 2-groups with precisely 4 cyclic subgroups of order 2n > 2, (f) new proofs of Blackburn's theorem on minimum nonmetacyclic teams, (g) class of p-groups all of whose subgroups of index p² are abelian, (h) category of 2-groups all of whose minimum nonabelian subgroups have order eight, (i) p-groups with cyclic subgroups of index p² are labeled.

Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey

George Mackey was once a unprecedented mathematician of significant strength and imaginative and prescient. His profound contributions to illustration conception, harmonic research, ergodic conception, and mathematical physics left a wealthy legacy for researchers that keeps this day. This ebook is predicated on lectures offered at an AMS unique consultation held in January 2007 in New Orleans devoted to his reminiscence.

Extra resources for Rings and modules

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.

Download PDF sample

Rated 4.57 of 5 – based on 48 votes