Can field have isomorphic image as non fields Can fields have a non field as isomorphic image ??
Also
If F:r---> R and if R is a field then r also has to be a field. If F is isomorphic function .
 A: Every ring homomorphism having a field as domain must be injective because the kernel is an ideal and a field has no non-trivial ideals. Therefore, the homomorphism induces an isomorphism of the field onto its image and so the image is a field.
A: Any non zero ring homomorphism maps an invertible element to an invertible element. So it will map a field to a field (and actually it will map to an isomorphic field, since the kernel of the map is $\{0\}$ because it's an ideal of that field).
If $f: A \to B$ is an isomorphism we also have an isomorphism $f^{-1}: B \to A$, so either both $A$ and $B$ are field, or none of them are. 
A: A field is, by (one possible) definition, a set with two binary operations $+$ and $\times$ that satisfy certain axioms, and those axioms can be formulated purely in terms of $+$ and $\times$.
If we have two structures $(A,+_A,\times_A)$ and $(B,+_B,\times_B)$ then an isomorphism between them is a bijection $f:A\to B$ such that $f(a_1+_Aa_2)=f(a_1)+_B f(a_2)$ and $f(a_1\times_Aa_2)=f(a_1)\times_Bf(a_2)$ for all $a_1,a_2\in A$.
If such an isomorphism exists, then the sentences in the language of $+$ and $\times$ that are true in $A$ are exactly the ones that are true in $B$ and vice versa.
In particular, if the field axioms are true in $A$, then they are also true in $B$. And if the field axioms are true in $B$, then they are also true in $A$.
So anything that a field is isomorphic to is itself a field.
Anything that a commutative ring is isomorphic to is itself a commutative ring.
And so forth.
