0
$\begingroup$

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"?

$\endgroup$
2
$\begingroup$

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

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.