Abstract algebra PLEASE SHOW ALL STEPS THANK YOU 365 Let R

#Abstract algebra

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3.65 Let R be a euclidean ring with degree function 0. (i) Prove that a (1) S 0 (a) for all nonzero a E R. (ii) Prove that a nonzero u E R is a unit if and only if 0(u) 00 l)

Solution

Since a,bR and b0,
then there exist q,rR such that a=qb+r with r=0, or r0 and (r)<(b). Since ba, r0. Hence (r)<(b). Since abab and aqb, ar too. Therefore there exists an element cR such that r=ac. Since ac=r0, we have (a)(r)<(b).

(2) The necessity is already proven by egreg, so we prove the sufficiency. Let (u)=1. There exist q,rR such that 1=qu+r1 with r=0, or r0 and (r)<(u). Assume that r0. Then (r)=0.
Since r=1 r0, (1)(r)=0, so (1)=0.
As showed egreg, in this case RR is a field, a contradiction. So r=0. Then u is a unit.