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 e R"Which of the following is a wrong statement in the proof?Assume R/N is commutative. Then rs = sr for all r, sE R/Nthen rs – sr E NVr, s E R, (rs + N)- (sr + N) = N iff (rs – sr) + N = NV r,s E R,(r + N) (s + N) = (s + N) (r + N) iff rs + N = sr + NIf rs – sr E N, Vr, s ER then rs - sr + N = N

The objective is to state the wrong statement while proving the theorem The quotient ring R modulo…

Answered: In the proof of the statement The…

In the proof of the statement The quoteint ring R modulo an ideal N, RIN, is commutative iff 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

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 33E: Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition...

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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 e R"
Which of the following is a wrong statement in the proof?
Assume R/N is commutative. Then rs = sr for all r, sE R/N
then rs – sr E N
Vr, s E R, (rs + N)- (sr + N) = N iff (rs – sr) + N = N
V r,s E R,
(r + N) (s + N) = (s + N) (r + N) iff rs + N = sr + N
If rs – sr E N, Vr, s ER then rs - sr + N = N

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