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 |
À l'intérieur du livre
Résultats 1-5 sur 32
Page
... endomorphism σ on k k[x;δ] differential polynomial ring with respect to a derivation δ on k [G, G] commutator subgroup of the group G [R, R] additive subgroup of the ring R generated by all [a, b] = ab − ba f.g. finitely generated ACC ...
... endomorphism σ on k k[x;δ] differential polynomial ring with respect to a derivation δ on k [G, G] commutator subgroup of the group G [R, R] additive subgroup of the ring R generated by all [a, b] = ab − ba f.g. finitely generated ACC ...
Page 1
... endomorphism ring for n (identical) copies of a module (Exercise 20), and the basic facts pertaining to direct decompositions of a ring into 1-sided or 2-sided ideals (Exercises 7 and 8). Throughout these exercises, the word “ring ...
... endomorphism ring for n (identical) copies of a module (Exercise 20), and the basic facts pertaining to direct decompositions of a ring into 1-sided or 2-sided ideals (Exercises 7 and 8). Throughout these exercises, the word “ring ...
Page 7
... endomorphism of M is an automorphism. (1) Show that any noetherian module M is hopfian. (2) Show that the left regular module R R is hopfian iff R is Dedekindfinite.3 (3) Deduce from (1), (2) that any left noetherian ring R is Dedekind ...
... endomorphism of M is an automorphism. (1) Show that any noetherian module M is hopfian. (2) Show that the left regular module R R is hopfian iff R is Dedekindfinite.3 (3) Deduce from (1), (2) that any left noetherian ring R is Dedekind ...
Page 12
... endomorphisms on the left. Let εj : M → nM be the jth inclusion, and πi : nM → M be the ith projection. For any endomorphism F: nM → nM, let fij be the composition πi Fεj ∈ E. Define a map α: End R(nM) → Mn(E) by α(F)=(f ij ...
... endomorphisms on the left. Let εj : M → nM be the jth inclusion, and πi : nM → M be the ith projection. For any endomorphism F: nM → nM, let fij be the composition πi Fεj ∈ E. Define a map α: End R(nM) → Mn(E) by α(F)=(f ij ...
Page 18
... endomorphism of the module (R/A)R.) Now λ(r) is the zero endomorphism iff rR ⊆ A, that is, r∈ A. Since λ is a ring homomorphism, it induces a ring embedding ER(A) → End R(R/A). We finish by showing that this map is onto. Given φ ...
... endomorphism of the module (R/A)R.) Now λ(r) is the zero endomorphism iff rR ⊆ A, that is, r∈ A. Since λ is a ring homomorphism, it induces a ring embedding ER(A) → End R(R/A). We finish by showing that this map is onto. Given φ ...
Table des matières
Jacobson Radical Theory | 49 |
Introduction to Representation Theory | 99 |
Ordered Structures in Rings 247 | 246 |
Perfect and Semiperfect Rings 325 | 324 |
Name Index | 349 |
Autres éditions - Tout afficher
Expressions et termes fréquents
0-divisor 2-primal abelian algebra artinian ring assume automorphism commutative ring conjugate constructed contradiction cyclic Dedekind-finite defined direct product direct summand division ring domain element endomorphism equation Exercise exists fact field finite group finite-dimensional follows group G hence homomorphism hopfian idempotent identity implies indecomposable induction infinite integer inverse irreducible isomorphism J-semisimple Jacobson radical k-algebra kG-module left ideal left primitive Lemma Let G local ring Math maximal ideal maximal left ideal Mn(k Mn(R module multiplication Neumann regular ring nil ideal Nilº noetherian ring noncommutative nonzero polynomial prime ideal primitive rings proof prove R-module R/rad rad kG representation resp right ideal right R-module ring theory semilocal ring semiprime semisimple ring show that rad simple ring Solution stable range subdirect product subgroup submodule subring suffices to show surjective Theorem unit-regular von Neumann regular zero