Relationship between interest rate and rental price of capital
The equation is exactly the same,

$$r(t) = R(t) - \delta$$

Now, set $\delta =1$ per assumptions, to obtain

$$r(t) = R(t) - 1 \implies 1+r(t) = R(t)$$

which reasonably continues to say that the accepted rental rate of capital $R(t)$, covers the depreciation rate plus $r(t)$ which is also the net return to capital.