Let $R$ be a Noetherian commutative ring. Let $I,J\subset K(R)$ be fractional ideals where $K(R)$ is the total quotient ring. Define $I^{-1}:=\{s\in K(R) : sI\subset R\}.$ Further suppose that $I$ is invertible I.e. $I^{-1}I=R$. Then $I^{-1}J=\{s\in K(R) : sI\subset J\}.$ (LHS is a product as ideals.)
Is this true and why?
(If necessary, we can add one more assumption: $J$ is also invertible.)