I have some questions about localization by maximal ideals.
Let $A$ be commutative ring and $p ⊂ A$ be a prime ideal.
then $A_p$ is local ring with maximal ideal $m = pA_p$.
Then I have question that
(i) $m^2 = p^2A_p$ is it correct?
Nextly $R$ be commutative ring and $m$ be its maximal ideal.
Here I have question.
(ii) $m/m^2$ and $mR_m/m^2R_m$ are isomorphic.
(iii) $m/m^2$ and $mR_m/(mR_m)^2$ are isomorphic.
Are these correct?
Regarding (ii) and (iii), if possible, please state proofs.
This may be a silly question, but I would be grateful if you answer.(Also, I'm sorry for my poor English.)
