Subset of Principal ideal

Question

Just started learning a little bit about ring theory. Can anyone give me a hint (or counterexample if the statement isn´t true). Let be $P $ Principal ideal then every Ideal $P^{'}\subset P$ is also a Principal ideal. May be a stupid question to ask but I need to know.

Answer 1

The counterexample $P = (1)$ works in any non-PID domain.
Even if we suppose that $P$ is a proper principal ideal, there are counterexamples: just take $P=(xy)$ in $\mathbb{C}[x,y]$. This contains the ideal $(x^2y, xy^2)$ that is not principal.

Answer 2

(I'm sticking to commutative rings with identity since I have a hunch that is what was intended. The following can be adapted for noncommutative rings in general, though.)

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As already mentioned, every ring with identity is a principal ideal $P$ of itself, and if ideals contained in principal ideals were principal, this would mean that all ideals of every ring with identity are principal. This is not the case.

Restricting the question to proper principal ideals does not help: you can take any nonzero ring with identity $R$, any nonzero ring $S$ with an ideal that is not principal, and $P=\{0\}\times S$ is a proper principal ideal of $R\times S$ which contains ideals that are not principal.

Is there any condition for $P$ so that $P′$ is a principal ideal ?

I think you are asking for conditions in which ideals contained in principal ideals are principal. Considering what has already been said, it is completely clear that this condition characterizes principal ideal rings.

In fact, you should now be able to prove why an ideal contained in a principal ideal might possibly not even be finitely generated.