Question

Prove that c and d are associates in R if and only $\raisebox{1ex}{$\mathbf{c}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}$}\right.$if and $\raisebox{1ex}{$\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{c}$}\right.$.

Short answer

It has been proved that c and d are associates in R if and only if $\raisebox{1ex}{$\mathbf{c}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}$}\right.$ and $\raisebox{1ex}{$\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{c}$}\right.$ .

Step-by-step solution

Proof of if statement

Let c and d are associates in R.

Then, $\mathit{c}\mathbf{=}\mathit{d}\mathit{u}$ for a unit $\mathit{u}\mathbf{\in }\mathit{R}$.

This implies,$\mathit{d}\mathbf{=}\mathit{c}\mathit{u}$ .

Thus, $\raisebox{1ex}{$\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{c}$}\right.$ and $\raisebox{1ex}{$\mathbf{c}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}$}\right.$.

Proof of only if statement

Let$\raisebox{1ex}{$\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{c}$}\right.$and $\raisebox{1ex}{$\mathbf{c}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}$}\right.$.

So, $\mathit{c}\mathbf{=}\mathit{d}\mathit{x}$ and $\mathit{d}\mathbf{=}\mathit{c}\mathit{y}$for some $\mathit{x}\mathbf{,}\mathit{y}\mathbf{\in }\mathit{R}$.

If $\mathit{c}\mathbf{=}{\mathbf{0}}_{\mathbf{R}}$then, $\mathit{d}\mathbf{=}{\mathbf{0}}_{\mathbf{R}}$.

Let $\mathit{C}\mathbf{\ne }{\mathbf{0}}_{\mathbf{R}}$.

Then, $\mathit{c}\mathbf{=}\left(cy\right)\mathit{x}$

Since R is an integral domain, c can be cancelled from both sides.

Thus, ${\mathbf{1}}_{\mathbf{R}}\mathbf{=}\mathit{y}\mathit{x}$.

This implies y and x are units.

Thus, c and d are associates.

Conclusion

Thus, it can be concluded that c and d are associates in R if and only if $\raisebox{1ex}{$\mathbf{c}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}$}\right.$ and $\raisebox{1ex}{$\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{c}$}\right.$ .