Let $R$ be a ring and $x\in R$
i) $x$ is a unit
ii) $\bar{x}$ is a unit in $R/P$, where $P$ is a prime ideal
iii) $\frac{x}{1}$ is a unit in $R_P$, where $P$ is a prime ideal
i)$\implies$ ii)
if $x$ is a unit then $\exists x^{-1}\in R$. let $x=\bar{x}+p$ and $x^{-1}=\bar{x}^{-1}+q$, where $p, q\in P$, so $xx^{-1}=(\bar{x}+p)(\bar{x}^{-1}+q)=\bar{x}\bar{x}^{-1}+p\bar{x}+q\bar{x}^{-1}+pq=\bar{x}\bar{x}^{-1}=1$, therefore $\bar{x}^{-1}$ is the inverse of $\bar{x}$ in $R/P$. But I haven't really used the fact that the ideal is prime.
I can't seem to figure out how to do the other two directions, any help would be appreciated.
i)$\implies$ ii). $xy=1\implies (x+P)(y+P)=xy+P=1+P$
ii)$\implies$ i). Is not true: For $q$, prime, $\mathbb Z/q\mathbb Z$ is a field.
i)$\implies$ iii). Is trivial: $xy=1\implies (x/1)(y/1)=1/1.$
iii)$\implies$ i). Is not true: Consider Field of fractions