If $A$ is isomorphic to $B$ and $B$ is a field, then $A$ is a field? This is a follow-up question to Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal. In that question, we know that since $S$ is a field and $R/\ker(f)$ is isomorphic to $S$ then $R/\ker(f)$ is also a field.
More generally, can we loose the condition from being isomorphic to being homomorphic?
Also, suppose $A$ is isomorphic to something weaker than a field, say a group, or an integral domain, is it possible that $A$ is actually a field?
 A: One thing that seems to confuse you is that the terminology "A is isomorphic to B" does not mean anything until you specify as what they are isomorphic.
For instance, it could be the case that A and B are both groups and that there is a bijective map $\varphi:A\to B$ that is a morphism of groups. In that case, they are isomorphic as groups.  This implies that if $A$ has any special property that a group could have, for instance $A$ is abelian or $A$ has only elements of order 2, then B must share that same property.
Now if A and B are both rings and there exist a morphism of rings between A and B then they are isomorphic rings. This implies that any special ring-theoretical property that A may have, must be shared by B. Being a field is actually such a special ring theoretical property, it could be expressed by the statement
$$( \forall a\in A \setminus \{0\}) ( \exists a'\in A )( aa'=a'a=1).$$
so in that sense the answer to your question is yes. A more complete statement would be given by

If A is a field and B is a ring and they are isomorphic as rings, then B is a field.

A: If $f:A\stackrel\sim\rightarrow B$ is an isomorphism of rings with unity and A is a field, then also $B$ is a field.
This is an easy exercise:if $ab=1_A$ then $f(a)f(b)=1_B$, and so on
A: Let $h:A \to B$ be an isomorphism from $A$ to $B$ which implies that $h$ is $1-1$ and $Onto$ mapping and preserves the operation(s). So $A$ and $B$ are of the same sizes and having same behavior with respect to the operation(s).
So if either $A$ or $B$ is field, then the other is also a field.
