ideals in a Dedekind domain

Let $R$ be a Dedekind domain,and let $𝔞$ and $𝔟$ be ideals of $R$.Then there is an element $\omega$ and an ideal $𝔠$ of $R$ such that

$$𝔞𝔠=(\omega )$$

and

$$𝔟+𝔠=R.$$

This result was proved by Steinitz in 1911.