Let $C$ be a subring of $B$ which is again a subring of a commutative ring $A$ , also suppose all of $A,B,C$ are Noetherian and $A \cong C$ , then is it true that $A \cong B$ ? If the claim is not true then what happens if we assume all the rings $A,B,C$ are Artinian ?
( all rings and subrings are with unity , where for subrings the unity is inherited from the super-ring structure )
Here is a non-artinian example, but which is "easier" than the one in the comments. Consider a situation like $$\mathbb C[x^n] \subset R\subset \mathbb C[x]$$ the outer two rings being isomorphic (send $x\mapsto x^n$). Now, it is easy to find subrings in between that are not isomorphic, for example if $n=3$ we could take $$R = \mathbb C[x^2,x^3] \cong \mathbb C[z,w]/(z^3 - w^2),$$ which is the coordinate ring of a cusp.