Question

Let R be a Euclidean domain. Prove that

1. $\mathit{\delta }\mathbf{\left(}{\mathbf{I}}_{\mathbf{R}}\mathbf{\right)}\mathbf{\le }\mathit{\delta }\mathbf{\left(}\mathbf{a}\mathbf{\right)}$for all nonzero$\mathit{a}\mathbf{\in }\mathit{R}$ .
2. If a and b are associates, then $\mathit{\delta }\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{=}\mathit{\delta }\mathbf{\left(}\mathbf{b}\mathbf{\right)}$
3. Ifrole="math" localid="1654615441384" $\mathit{a}\mathbf{/}\mathit{b}$ and $\mathit{\delta }\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{=}\mathit{\delta }\mathbf{\left(}\mathbf{b}\mathbf{\right)}$, then a and b are associative.

Short answer

1. It is proved that$\delta \left(I_R\right)\le \delta \left(a\right)$ for all nonzero $a\in R$.
2. It is proved that If a and b are associates, then$\delta \left(a\right)=\delta \left(b\right)$
3. It is proved that If role="math" localid="1654615587291" $a/b$and $\delta \left(a\right)=\delta \left(b\right)$, then a and b are associative.

Step-by-step solution

(a) Show δ(IR)≤δ(a) for all non-zero a∈R

Now, by definition of Euclidean domain, An integral domain R is said to be Euclidean domain if there is a function $\mathbf{\delta }$from the non-zero elements of R to non negative integers with these properties:

1. If role="math" localid="1654616111369" $\mathbf{a}$are role="math" localid="1654616114321" $\mathbf{b}$non-negative elements of R, then role="math" localid="1654616118012" $\mathbf{\delta }\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{\le }\mathbf{\delta }\mathbf{\left(}\mathbf{ab}\mathbf{\right)}$.
2. If role="math" localid="1654616125325" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{R}$and role="math" localid="1654616122237" $\mathit{b}\mathbf{\ne }{\mathbf{0}}_{\mathbf{R}}$, then there exist role="math" localid="1654615914379" $\mathit{q}\mathbf{,}\mathit{r}\mathbf{\in }\mathit{R}$such that role="math" localid="1654616129764" $\mathit{a}\mathbf{=}\mathit{b}\mathit{q}\mathbf{+}\mathit{r}$and either $\mathit{r}\mathbf{=}{\mathbf{0}}_{\mathbf{R}}$or $\mathbf{\delta }\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{\le }\mathbf{\delta }\mathbf{\left(}\mathbf{b}\mathbf{\right)}$.

Now,$R$ is an integral domain

Therefore,$I_R\in R$ and for $a\in R$

By definition, we can write

$\begin{array}{l}\delta \left(I_R\right)\le \delta \left(I_{R}a\right)=\delta \left(a\right)\\ \Rightarrow \delta \left(I_R\right)\le \delta \left(a\right)\end{array}$

Hence proved.

(b) Show If a and b are associates then  δ(a)=δ(b)

By using definition of associate, we can say that Let R be a commutative ring with unit element. Two elements a and b in R is said to be associate if $\mathbf{b}\mathbf{=}\mathbf{au}$ where u is unit in R.

Also, we know that a is associate of b if and only if b is associate of a.

$\begin{array}{l}\Rightarrow a=bu\text{where}u\text{isunitin}R\\ \Rightarrow b=au\text{wherevisunitin}R\end{array}$

By exercise 17(b) We know that if $\mathrm{R}$ is a Euclidean domain, $\mathit{\delta }\mathbf{\left(}\mathbf{a}\mathbf{b}\mathbf{\right)}\mathbf{=}\mathit{\delta }\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathit{\delta }\mathbf{\left(}\mathbf{b}\mathbf{\right)}$

Therefore,

$\begin{array}{l}\Rightarrow \delta \left(ab\right)=\delta \left(bu\right)\delta \left(av\right)=\delta \left(b\right)\delta \left(u\right)\delta \left(a\right)\delta \left(v\right)\\ =\delta \left(b\right)\delta \left(a\right)[\because \delta \left(u\right),\delta \left(v\right)=1]\\ =\delta \left(a\right)\delta \left(b\right)\\ \Rightarrow \delta \left(ab\right)=\delta \left(b\right)\delta \left(a\right)\end{array}$

It is proved that If a and b are associates, then $\mathrm{\delta }\left(\mathrm{a}\right)=\mathrm{\delta }\left(\mathrm{b}\right)$.

(c)  Show If a/b and δ(a)=δ(b) then a and b are associate

Now, we have $a/b\Rightarrow b=ac$

$\Rightarrow \delta \left(b\right)=\delta \left(ap\right)$

By exercise 17(b) We know that if $R$is a Euclidean domain, $\mathrm{\delta }\left(\mathrm{ab}\right)=\mathrm{\delta }\left(\mathrm{a}\right)\mathrm{\delta }\left(\mathrm{b}\right)$

$\Rightarrow \delta \left(b\right)=\delta \left(a\right)\delta \left(p\right)$

Also, we have localid="1654616770425" $\delta \left(a\right)=\delta \left(b\right)$

$\Rightarrow \delta \left(p\right)=1$

$\Rightarrow \delta \left(p\right)=1$

$\Rightarrow p$is a unit.

$\Rightarrow b=ap\text{Where}p\text{isaunit}$.

Therefore, a and b are associate.