New PDF release: Algebras of Linear Transformations

By Douglas R. Farenick

ISBN-10: 0387950621

ISBN-13: 9780387950624

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K-1, j = 0, ... , m-1, respectively, have the O desired properties. 5 (Jordan-Holder) If, in a module M, there exists a composition chain, then every chain of submodules of M can be refined so as to become a composition chain. The lengths of all composition chains of the module M are equal. 6 If, in a module M, there exists a composition chain, its length is called the length of the module M, and denoted by /(M). If there are no composition chains in M, we write /(M) = oo. Thus the number /(M) is the upper bound of the lengths of chains of submodules of M.

3. S = 0, then Rs/PRs ~ (R/P),cs>o where v:R-+ R/P is the natural homomorphism. Deduce that if Q => P is also a prime ideal then R 0 /PR 0 ~ (R/P) 01p. 4. If t is a non-nilpotent element of a ring R, write R, for the ring of fractions of R with respect to the multiplicative set {l,t,t 2 , ... },and denote by S the set {reR:r~Ker(R ~ R,) => Ann(r)}. (i) Prove that Sis the inverse image of the set of non-zero-divisors by the homomorphism R-+ R,. (ii) Prove that the total ring of fractions of R, is isomorphic to Rs.

5 contains the very useful Artin-Rees Lemma and its consequences, including Krull Intersection Theorem. 6, devoted to completions. Considering topologies of a special type of rings and modules, we study the operation of completion, defined in terms of Cauchy sequences and known from elementary analysis and topology. Complete rings and modules have many remarkable properties. 6 we concentrate upon the so-called adic topologies determined by the powers of a fixed ideal. 7, the closing section of Chapter II, deals with Artin rings, which may be defined dually to Noetherian rings by means of the descending chain condition for ideals.

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Algebras of Linear Transformations by Douglas R. Farenick

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