Question

Let Rbe a Euclidean domain and u$\mathbf{\in }$R. Prove that u is a unit if and only if $\mathit{\delta }\mathbf{\left(}\mathit{u}\mathbf{\right)}\mathbf{=}\mathit{\delta }\mathbf{\left(}{\mathbf{1}}_{\mathbf{R}}\mathbf{\right)}$.

Short answer

It has been proved that in a Euclidean domain R,u is a unit if and only if $\delta \left(u\right)=\delta \left(1_R\right)$.

Step-by-step solution

Given that

Given thatR is a Euclidean domain and$u\in R$ .

Suppose that u is unit

Firstly, suppose that u is a unit and prove that $\delta \left(u\right)=\delta \left(1_R\right)$.

For any nonzero it is certain that $\delta \left(1_R\right)\le \delta \left(1_{R}a\right)=\delta \left(a\right)$.…..(1)

If u is a unit, then there exists v such that $uv=1_R$.

Then $\delta \left(u\right)\le \delta \left(uv\right)=\delta \left(1_R\right)$ ……(2)

From (1) and (2)

It is clear that $\delta \left(u\right)=\delta \left(1_R\right)$.

Suppose thatδ(u)=δ(1R).

Conversely suppose that $\delta \left(u\right)=\delta \left(1_R\right)$.

Since R is a Euclidean domain and u is a non zero element in R, using theorem 10.2

$\delta \left(u\right)=\delta \left(1_R\right)$implies that u is a unit.

Conclusion

Thus, it can be concluded thatu is a unit if and only if $\delta \left(u\right)=\delta \left(1_R\right)$.