Exercises in Classical Ring TheorySpringer Science & Business Media, 9 mai 2006 - 364 pages This useful book, which grew out of the author's lectures at Berkeley, presents some 400 exercises of varying degrees of difficulty in classical ring theory, together with complete solutions, background information, historical commentary, bibliographic details, and indications of possible improvements or generalizations. The book should be especially helpful to graduate students as a model of the problem-solving process and an illustration of the applications of different theorems in ring theory. The author also discusses "the folklore of the subject: the `tricks of the trade' in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference". The problems are from the following areas: the Wedderburn-Artin theory of semisimple rings, the Jacobson radical, representation theory of groups and algebras, (semi)prime rings, (semi)primitive rings, division rings, ordered rings, (semi)local rings, the theory of idempotents, and (semi)perfect rings. Problems in the areas of module theory, category theory, and rings of quotients are not included, since they will appear in a later book. T. W. Hungerford, Mathematical Reviews |
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Résultats 1-5 sur 33
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... subring of R means a subring containing the identity of R ( unless otherwise specified ) . The word “ ideal ” always means a two - sided ideal ; an adjective such as " noetherian " likewise means both right and left noetherian . A ...
... subring of R means a subring containing the identity of R ( unless otherwise specified ) . The word “ ideal ” always means a two - sided ideal ; an adjective such as " noetherian " likewise means both right and left noetherian . A ...
Page 1
... subring” always means a subring containing the identity element of the larger ring. If R = {0}, R is called the zero ring; note that this is the case iff 1 = 0 in R. If R = {0} and ab = 0⇒ a = 0 or b = 0, R is said to be a domain ...
... subring” always means a subring containing the identity element of the larger ring. If R = {0}, R is called the zero ring; note that this is the case iff 1 = 0 in R. If R = {0} and ab = 0⇒ a = 0 or b = 0, R is said to be a domain ...
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... subring of M2 ( k ) , and that it is isomorphic to the ring R of 2 x 2 lower triangular matrices over k . 1 Solution . Let a = ( 64 ) Then a -1Ra consists of the matrices y x y z + z - ( 67 ) ( 9 ) ( 1 ) = ( ~~ " " 77 " ) . 1 ) y У We ...
... subring of M2 ( k ) , and that it is isomorphic to the ring R of 2 x 2 lower triangular matrices over k . 1 Solution . Let a = ( 64 ) Then a -1Ra consists of the matrices y x y z + z - ( 67 ) ( 9 ) ( 1 ) = ( ~~ " " 77 " ) . 1 ) y У We ...
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... subring of R. Since A AC A , A CIR ( A ) . Clearly A is an ideal in IR ( A ) . Conversely , if A is an ideal in some subring SCR , then rЄ S implies rA C A , so rЄ IR ( A ) . This shows that SC IR ( A ) . ( 2 ) Define : IR ( A ) → EndR ...
... subring of R. Since A AC A , A CIR ( A ) . Clearly A is an ideal in IR ( A ) . Conversely , if A is an ideal in some subring SCR , then rЄ S implies rA C A , so rЄ IR ( A ) . This shows that SC IR ( A ) . ( 2 ) Define : IR ( A ) → EndR ...
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... subring of End ( c ( R / A ) ) . Therefore , we have ( 5 ) ⇒ ( 1 ) Next , suppose ( 6 ) holds . Then c ( R / A ) can be identified with C / I for some ideal I of C. Then 21 ༤ End ( c ( R / A ) ) ≈ End ( c ( C / I ) ) ≈ End ( c / 1 ...
... subring of End ( c ( R / A ) ) . Therefore , we have ( 5 ) ⇒ ( 1 ) Next , suppose ( 6 ) holds . Then c ( R / A ) can be identified with C / I for some ideal I of C. Then 21 ༤ End ( c ( R / A ) ) ≈ End ( c ( C / I ) ) ≈ End ( c / 1 ...
Table des matières
2 | 49 |
6 Group Rings and the JSemisimplicity Problem | 80 |
Introduction to Representation Theory | 132 |
Prime and Primitive Rings | 141 |
3 | 157 |
11 Structure of Primitive Rings the Density Theorem | 161 |
26 | 171 |
32 | 191 |
15 Tensor Products and Maximal Subfields | 228 |
Ordered Structures in Rings 247 | 246 |
17 Orderings and Preorderings in Rings 247 | 253 |
Local Rings Semilocal Rings and Idempotents | 268 |
Perfect and Semiperfect Rings 325 | 326 |
24 Homological Characterizations of Perfect and Semiperfect | 336 |
Name Index | 349 |
49 | 350 |
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Expressions et termes fréquents
0-divisor 2-primal abelian artinian ring assume central idempotents commutative ring conjugate constructed contradiction cyclic defined direct product division algebra division ring domain element endomorphism equation Exercise exists fact FC-p field finite finite-dimensional follows group G hence homomorphism idempotent identity implies indecomposable induction integer inverse isomorphism J-semisimple Jacobson k-algebra kG-module left ideal left primitive ring Lemma local ring Math matrix ring maximal ideal maximal left ideal maximal subfield Mn(R module multiplication Neumann regular ring nil ideal nilpotent ideal noetherian ring noncommutative polynomial prime ideal primitive idempotents primitive rings projective proof prove R-module R/rad radical resp right ideal right perfect right R-module ring theory semilocal ring semiprimary ring semisimple ring simple ring Solution stable range strongly regular subdirect product subdirectly irreducible subgroup submodule subring suffices to show Theorem unit-regular von Neumann regular zero