a) LetRbe a ring. We define the radical ofRto be the left idealNwhich is the intersection of all maximal left ideals ofR. Show thatNE=0for every simpleR-moduleE. Show thatNis a two-sided ideal. (b) Show that the radical ofR/Nis 0 .

a) LetRbe a ring. We define the radical ofRto be the left idealNwhich is the intersection of all maximal left ideals ofR. Show thatNE=0for every simpleR-moduleE. Show thatNis a two-sided ideal. (b) Show that the radical ofR/Nis 0 .

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 33E: 33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all...

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1. (a) LetRbe a ring. We define the radical ofRto be the left idealNwhich is the intersection of all maximal left ideals ofR. Show thatNE=0for every simpleR-moduleE. Show thatNis a two-sided ideal. (b) Show that the radical ofR/Nis 0 .

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