It's maybe stupid question but I want to be sure about what I've learned.
Is the quotient $R$-module $R/(p^n) \simeq R/(p) \oplus...\oplus R/(p)$ in general, if not when?
I think over $\mathbb{Z}$-module $\mathbb{Z}/p^2$ and $\mathbb{Z}/p \oplus \mathbb{Z}/p $. And they are not isomorphic since multiplying to p, on one hand I got zero, on the other hand I did not. Am I correct?
Thank you.