Topic Abstract algebra Question Let R be a euclidean ring an

Topic: Abstract algebra

Question: Let R be a euclidean ring and let a be an element of R be non zero.
Proof that if d(a)= d(1), then a is a unit in R

Please show all steps
Thank you kindly :)

Solution

Assume that your d function satisfies d(x)d(z) if x divides z. (BTW, this is the same as saying that d(x)d(xy))

Since 1 divides every element, we have d(1)d(x) for all x. If a is a unit, then a divides 1 and so d(a)d(1).

This implies d(a)=d(1)

Conversely, as you have remarked, 1=aq+r with r=0 or d(r)<d(a).

But if d(a)=d(1), then if r0 we would get d(r)<d(1), which contradicts d(1)d(x) for all x.

Hence r=0 and a is a unit.