Prove that there is a unit $u \in R$ such that $ub = bu = a$
We know that $b^3 = b$ and so $b^4 = b^2$. We may then compute that $$ (b^2+b-1)^2 = b^4 + 2 b^3 - b^2 - 2 b + 1 = b^2 + 2b - b^2 - 2b + 1 = 1. $$ Hence $b^2+b-1$ is a unit. Furthermore, we have $$ (b^2+b-1)b = b(b^2+b-1) = b^3 + b^2 - b = b + b^2 - b = b^2, $$ so it is the unit we are after.