In the proof of the statement" The quoteint ring R modulo an ideal N, R/N, is commutative iffrs – sr e N for all r, s eR"Which of the following is a wrong statement in the proof?Vr, s E R, (r+ N) (s + N) = (s + N) (r + N) iff rs + N = sr + NVr, s E R, (rs + N) – (sr + N) = N iff (rs – sr) + N = NAssume R/N is commutative. Then rs = sr for all r, s E RINthen rs – sr e NIf rs – sr e N, Vr, s ER then rs – sr + N = N

The quotient ring R modulo an ideal N , R\N is commutative iff rs-sr∈N

Answered: In the proof of the statement " The…

In the proof of the statement " The quoteint ring R modulo an ideal N, R/N, is commu rs - sr E N for all r, s eR" Which of the following is a wrong statement in the proof?

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 30E: a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R....

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Question

In the proof of the statement
" The quoteint ring R modulo an ideal N, R/N, is commutative iff
rs – sr e N for all r, s eR"
Which of the following is a wrong statement in the proof?
Vr, s E R, (r+ N) (s + N) = (s + N) (r + N) iff rs + N = sr + N
Vr, s E R, (rs + N) – (sr + N) = N iff (rs – sr) + N = N
Assume R/N is commutative. Then rs = sr for all r, s E RIN
then rs – sr e N
If rs – sr e N, Vr, s ER then rs – sr + N = N

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