Exercises in Classical Ring TheorySpringer Science & Business Media, 29 juin 2013 - 288 pages Based in large part on the comprehensive "First Course in Ring Theory" by the same author, this book provides a comprehensive set of problems and solutions in ring theory that will serve not only as a teaching aid to instructors using that book, but also for students, who will see how ring theory theorems are applied to solving ring-theoretic problems and how good proofs are written. The author demonstrates that problem-solving is a lively process: in "Comments" following many solutions he discusses what happens if a hypothesis is removed, whether the exercise can be further generalized, what would be a concrete example for the exercise, and so forth. The book is thus much more than a solution manual. |
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Résultats 1-5 sur 75
Page 1
... isomorphisms of modules and rings , the chain conditions , and Dedekind - finiteness . A ring R is said to be Dedekind - finite if ab = 1 in R implies that ba = 1. The chain conditions are the usual noetherian ( ACC ) or artinian ( DCC ) ...
... isomorphisms of modules and rings , the chain conditions , and Dedekind - finiteness . A ring R is said to be Dedekind - finite if ab = 1 in R implies that ba = 1. The chain conditions are the usual noetherian ( ACC ) or artinian ( DCC ) ...
Page 4
... isomorphism between R and the direct product of rings B1 x x Bn . Show that any isomorphism of B1 × R with a finite direct product of rings arises in this way . ... Solution . Suppose e2's are idempotents with the given properties , and ...
... isomorphism between R and the direct product of rings B1 x x Bn . Show that any isomorphism of B1 × R with a finite direct product of rings arises in this way . ... Solution . Suppose e2's are idempotents with the given properties , and ...
Page 5
... isomorphism of rings . The last part of the Exercise is now routine : we omit it . ... Ex . 1.8 . Let R = B1 → Bn , where the B ; ' s are ideals of R. Show that any left ideal ( resp . ideal ) I of R has the form I = I1 ... In where ...
... isomorphism of rings . The last part of the Exercise is now routine : we omit it . ... Ex . 1.8 . Let R = B1 → Bn , where the B ; ' s are ideals of R. Show that any left ideal ( resp . ideal ) I of R has the form I = I1 ... In where ...
Page 6
... isomorphism from S to R. Note that S is also isomorphic to the " subrng " of M2 ( k ) consisting of matrices of the form isomorphism given by ( a , b ) → ( ( a ) . a b a b :) with an Ex . 1.11 . Let R be a ring possibly without an ...
... isomorphism from S to R. Note that S is also isomorphic to the " subrng " of M2 ( k ) consisting of matrices of the form isomorphism given by ( a , b ) → ( ( a ) . a b a b :) with an Ex . 1.11 . Let R be a ring possibly without an ...
Page 9
... isomorphic to H , the division ring of real quaternions . ( c ) Show that A / A ( x2 + 1 ) is isomorphic to M2 ( C ) . • Solution . ( a ) Here A is the ring of all " skew polynomials ” Σa ; x1 ( a ; ← C ) , multiplied according to the ...
... isomorphic to H , the division ring of real quaternions . ( c ) Show that A / A ( x2 + 1 ) is isomorphic to M2 ( C ) . • Solution . ( a ) Here A is the ring of all " skew polynomials ” Σa ; x1 ( a ; ← C ) , multiplied according to the ...
Table des matières
1 | |
16 | |
Jacobson Radical Theory 4 The Jacobson radical 123535 | 35 |
5 Jacobson radical under change of rings | 52 |
Introduction to Representation Theory 689 | 69 |
Linear groups | 98 |
Prime and Primitive Rings 103 | 102 |
11 Structure of primitive rings the Density Theorem | 119 |
Introduction to Division Rings | 151 |
15 Tensor products and maximal subfields | 176 |
Ordered Structures in Rings | 191 |
Local Rings Semilocal Rings | 211 |
21 The theory of idempotents | 229 |
22 Central idempotents and block decompositions | 251 |
24 Homological characterizations of perfect | 265 |
Subject Index | 281 |
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Expressions et termes fréquents
a₁ abelian artinian ring assume automorphism B₁ central idempotents char commutative ring conjugate constructed contradiction decomposition Dedekind-finite defined division ring domain element endomorphism equation Exercise exists fact finite group finite-dimensional follows group G hopfian idempotent identity implies indecomposable integer inverse irreducible isomorphism J-semisimple Jacobson radical k-algebra kG-module left ideal left primitive ring Lemma Let G linear local ring matrix maximal ideal maximal left ideal maximal subfield minimal left Mn(R module multiplication Neumann regular ring nil ideal nilpotent ideal noetherian ring noncommutative nonzero polynomial prime ideal primitive idempotents primitive rings proof prove quasi-regular R-module R/rad rad kG representation resp right ideal right R-module ring theory semilocal ring semiprime semisimple ring show that rad simple left R-module simple ring soc(RR Solution stable range strongly regular subdirect product subgroup subring Theorem unit-regular zero