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?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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
Transcribed Image Text: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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