Suppose $f: A \to B$ and $g: B \to A$ are injections of rings
(*commutative with identity*). Must $A$ and $B$ be isomorphic as
rings?


According to [this
question](https://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold), this answer should be "no", but can someone give an example?


Thanks!



Schroeder-Bernstein for Rings
Hey Damien, I think the following should work $\mathbb{C}$ and $\mathbb{C}(x)$. There is only one uncountable algebraically closed field of each cardinality in characteristic 0 and the algebraic closure of the right hand guy should have cardinality the continuum it should be isomorphic to $\mathbb{C}$. Probably, this assumes the axiom of choice though.


