Download e-book for kindle: A First Course in Abstract Algebra by Hiram Paley

By Hiram Paley

ISBN-10: 0030549655

ISBN-13: 9780030549656

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Comme le B-module à droite Bd est plat, il s u f i t d'appliquer la prop. 8. COROLLAIRE 3. - Soient R, S deux anneaux, rp : R -t S un homomorphisme d'anneaux. Si M est un S-module à droite plut et s i q*(Sd) est un R-module à droite plat, alors

3) On considère un diagramme commutatif de groupes commutatifs V' OU' = O dans lequel on suppose que les lignes et les colonnes sont exactes, que d et u" sont injectifs et a" surjectif. Montrer que dans ces conditions u"' est injectif. Généraliser. 4 l 59 EXERCICES 4) On considère un diagramme commutatif de groupes commutatifs où on suppose que les deux lignes sont exactes. a) Montrer que, si a est surjectif, b et d injectifs, alors c est injectif. b) Montrer que, si d est injectif, a et c surjectifs, alors b est surjectif.

VII, $ 2, exerc. 7. b) Soient M un A-module à droite, M' un sous-module de M, M" un sous-module de M'. Montrer que, si M' est un sous-module pur de M et M" un sous-module pur de M', alors M" est un sous-module pur de M, et M'/Mu est un sous-module pur de M/M". Si M" est un sous-module pur de M, M" est sous-module pur de M'. c) Montrer que, si N et P sont deux sous-modules de M tels que N n P et N P soient purs dans M, alors N et P sont des sous-modules purs de M. Donner un exemple de deux sous-modules N, P de Z2qui sont purs dans Z2 mais tels que N P ne soit pas un sous-module pur de Z2.

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A First Course in Abstract Algebra by Hiram Paley

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