I was studying commutative algebra and I have found several problems and results involving powers of ideals. However, I am not very good at working with those objects and I have found some dificulties.
If $R$ is a ring and $I$ is an ideal of $R$ we can consider the quotient $R\left/I\right.$. Now let $J$ be an ideal of $R$ such that $I\subset J$, then $J\left/I\right.$ is an ideal of $R\left/I\right.$. What can we say abut $\left(J\left/I\right.\right)^n$? Is it true that $\left(J\left/I\right.\right)^n=J^n\left/I\right.$? If not, can we add some conditions on $J$ or $I$ such that this holds? (prime, maximal, principal,...)
So any help will be mostly appreciated, and thanks in advance.
It is not necessarily true, simply because we do not always have $I\subset J^n$. What can be said is that $$\bigl(J/I\bigr)^n=\bigl(J^n+I\bigr)/I\simeq J^n/\bigl(J^n\cap I\bigr).$$