TITLE: Steps to prove or disprove if two rings are isomorphic
QUESTION [5 upvotes]: So i'm struggling on how to prove if two rings are not isomorphic to one another. My professor told me that if a ring is not isomorphic to another, the best way to prove that this is true is to find a preserved property of isomorphisms that is not held.
So i considered the following:
1.Q and R (quotients and rationals)
2.$~~\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/16\mathbb{Z}$ (Z mod 4 cross Z mod 4 and Z mod 16)
I cannot seem to think of any of the properties:
communicative, identity, integral domain, and field property that do not hold for rings. My professor told me this it isnt enough to give an example mapping like F: Q -> R and show that it isnt isomorphic.
Hence, how can i show that these two problems above arent isomorphic?
Can anyone give me some finite steps to prove is something is isomorphic to something or not?

REPLY [3 votes]: 1.
The rationals $\mathbb Q$ and the reals $\mathbb R$ are not isomorphic because they don't have the same cardinality.
If you want an algebraic reason, there is an element $u \in \mathbb R$ such that $u^2=2$ but there is not such element in $\mathbb Q$.
In fact, every ring homomorphism $\mathbb Q \to \mathbb R$ must be the inclusion, and so cannot be surjective.
2.
Consider the idempotents in each ring, that is the solutions of $x^2=x$. 
The idempotents of $\mathbb Z_{16}$ are $0$ and $1$ but there are at least $4$ idempotents in $\mathbb Z_{4}\times \mathbb Z_{4}$, namely $(0,0), (1,0), (0,1), (1,1)$.