If a principal ideal is a multiple of another ideal, is it necessarily a principal multiple?

Question

Question. Let $R$ denote a commutative ring. Suppose $\frak{a}$ and $\frak{b}$ are principal ideals of $R$. Suppose there exists a (not-necessarily principal) ideal $\frak{j}$ of $R$ satisfying $\frak{ja=b}.$ Does there necessarily exist a principal ideal $\frak{j}$ such that $\frak{ja=b}$? If the answer is "no", are there assumptions on $R$ weaker than being a principal ideal ring that make the answer "yes"?

Answer 1

Let $\mathfrak a = (a)$ and $\mathfrak b = (b)$.

$\mathfrak b = \mathfrak j \mathfrak a \subseteq \mathfrak a$, so $b = ka$ for some $k$.

Therefore, $\mathfrak b = (b) = (ka) = (k) (a) = (k) \mathfrak a$.