For localizationg of a ring in a prime number (noted p), I mean (that's how my book defines it) all the elements of the field of fractions of the ring in question of the form $\frac{a}{b}$ such that b can't be divided by p. In the case of $\mathbb{Z}$ and p, we are talking of {$\frac{a}{b} \in \mathbb{Q}$ such that p doesn't divide b}.
With this in mind, are the localization of $\mathbb{Z}$ in 2, $\mathbb{Z}_{(2)}$, and the localization of $\mathbb{Z}$ in 3, $\mathbb{Z}_{(3)}$, isomorphic? Can you explain well how?
I was thinking of proceeding using the ideals of these rings pˆn$\mathbb{Z}_{(p)}$, I am not sure how, but they should be bilateral ideals and therefore the kernel of some homomorphism. I was thinking of using the universal property of the quotient or the correspondence theorem, but didn't get far. I would include more work, but it is almost non-existent.
There residue fields are not isomorphic, because of $$ \Bbb Z_{(p)}/(p)\cong \Bbb F_p. $$ Hence the $p$-local numbers are not isomorphic as rings for different primes.