Inverse of integral ideal in a Dedekind domain is a specific fractional ideal

Question

Let $R$ be a Dedekind domain with fraction field $k$. Incan show that if $p$ is a prime ideal, then $p^{-1}=\{x\in k \mid xp\subset R \}$ satisfies $pp^{-1}=R$. Moreover, if $I$ is any integral ideal of R, then we can factorize it into primes $$I=p_1\cdots p_r$$ And its inverse is $p_1^{-1}\cdots p_r^{-1}$. Does this inverse coincide with the fractional ideal $I^{-1}=\{ x\in k \mid xI\subset R\}$?