Consider the following assertions about a commutative ring $R$ with identity and elements $a, b \in R$ :
$\text{(I)}$ There exist $p, q \in R$ such that $a p+b q=1$.
$\text{(II)}$ There exist $p, q \in R$ such that $a^{2} p+b^{2} q=1$.
Then:
- $\text{(I)}$ implies $\text{(II)}$, and $\text{(II)}$ implies $\text{(I)}$.
- $\text{(I)}$ implies $\text{(II)}$, but $\text{(II)}$ does not imply $\text{(I)}$.
- $\text{(I)}$ does not imply $\text{(II)}$, but $\text{(II)}$ implies $\text{(I)}$.
- $\text{(I)}$ does not imply $\text{(II)}$, and $\text{(II)}$ does not imply $\text{(I)}$.
