According to Wikipedia (and some other resources), if every ideal in a ring (or domain) $R$ can be decomposed into a product of prime ideals, then the factorization is unique up to the order of factors.
Is there a direct proof of that? (i.e. Do not prove $R$ is Dedekind first.)
From the proposition
If $\mathfrak{p}$ is prime and $\mathfrak{p}\supseteq\mathfrak{a}\mathfrak{b}$, then $\mathfrak{p}\supseteq\mathfrak{a}$ or $\mathfrak{p}\supseteq\mathfrak{b}$.
we know that $\mathfrak{p}$ in one factorization contains some prime factor $\mathfrak{q}$ in another factorization. This does not show that $\mathfrak{p}=\mathfrak{q}$ since we have not proved the Krull dimension is $1$.
If this is the case, we still need to show that the correspondent exponents are equal.
(We always assume that rings are commutative and have identity element $1\ne 0$.)
