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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... follows immediately from ( b ) . Ex . 1.4 * . Let a Є R. ( 1 ) Show that if a has a left inverse , then a is not a left 0 - divisor . ( 2 ) Show that the converse holds if a Є aRa . Solution . ( 1 ) Say ba = 1. Then ac = 0 implies c ...
... follows immediately from ( b ) . Ex . 1.4 * . Let a Є R. ( 1 ) Show that if a has a left inverse , then a is not a left 0 - divisor . ( 2 ) Show that the converse holds if a Є aRa . Solution . ( 1 ) Say ba = 1. Then ac = 0 implies c ...
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... follows by combining the " left - invertible " and " right - invertible " cases . - Comment . The formula for ( 1 — ab ) −1 above occurs often in linear algebra books ( for n x n matrices ) . Kaplansky taught me a way in which you can ...
... follows by combining the " left - invertible " and " right - invertible " cases . - Comment . The formula for ( 1 — ab ) −1 above occurs often in linear algebra books ( for n x n matrices ) . Kaplansky taught me a way in which you can ...
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... follows that is an isomorphism . ( Note that here is not a homomorphism of left C - vector spaces , since is not the identity map from CCA to C C M2 ( C ) ! ) Ex . 1.16 . Let K be a division ring with center k . ( 1 ) Show that the ...
... follows that is an isomorphism . ( Note that here is not a homomorphism of left C - vector spaces , since is not the identity map from CCA to C C M2 ( C ) ! ) Ex . 1.16 . Let K be a division ring with center k . ( 1 ) Show that the ...
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... follows that h € Z ( R ) = k [ x ] . • Ex . 1.17 . Let x , y be elements in a ring R such that Rx = Ry . Show that there exists a right R - module isomorphism ƒ xRyR such that f ( x ) = y . : Solution . Define ƒ ( xr ) = yr for any rЄ R ...
... follows that h € Z ( R ) = k [ x ] . • Ex . 1.17 . Let x , y be elements in a ring R such that Rx = Ry . Show that there exists a right R - module isomorphism ƒ xRyR such that f ( x ) = y . : Solution . Define ƒ ( xr ) = yr for any rЄ R ...
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... . involutions ) on the subrings R and S of A. ( 3 ) follows immediately from ( 1 ) and ( 2 ) . ( Conversely , one can also show that RR ° Pk k ° P . ) Comment . In ( 2 ) above , a and §1 . Basic Terminology and Examples 13.
... . involutions ) on the subrings R and S of A. ( 3 ) follows immediately from ( 1 ) and ( 2 ) . ( Conversely , one can also show that RR ° Pk k ° P . ) Comment . In ( 2 ) above , a and §1 . Basic Terminology and Examples 13.
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