About isomorphism of rings and fields If $A,B$ are rings and $A$ is a field. If $A$ is a field and $A\cong B$ so $B$ is a field too?
Thank you!

Suppose $\phi: A \leftrightarrow B$ is a ring isomorphism. $B$ is a field if it is commutative, and if every non-zero element of $B$ has a multiplicative inverse.
Let $a,b$ be two elements of $B$. Then
$$ab = \phi\left(\phi^{-1}(a)\right)\phi\left(\phi^{-1}(b)\right) = \phi\left(\phi^{-1}(a)\phi^{-1}(b)\right) = \ldots$$
can you finish the argument?
Similarly, let $b$ be a nonzero element of $B$. Can you prove $\phi^{-1}(b)$ is nonzero? How might you use $\phi^{-1}(b)$ to construct the multiplicative inverse of $b$?
