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Sudhanshu Ranjan posted an Question
August 16, 2020 • 06:49 am
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30 points
30 points
- CSIR NET
- Mathematical Sciences
Suppose r is an integral domain which is not a field. show that if r is euclidean domain, then r has universal side divisiors
Suppose R is an integral domain which is not a field. Show that if R is Euclidean domain, then R has universal side divisiors
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Shashi ranjan sinha![best-answer]()
By Euclidean domain definition, we can write x =qu + r, where r= 0 or N(r) < N(u). If r= 0, we are done. so let r be non zero. since u has minimal norm among all non zero & non unit elements, therefore N(r) < N(u) implies r must be either zero or a unit. but we assume in this case that r is non zero. so we are left with r to be unit. Thus, x = qu+r, where either r= o or r is unit. hence by definition, u is universal side divisior