I asked this question on Mathematics Stackexchange, but got no answer.
Let $A$ and $B$ be noetherian commutative rings with one, and let $f:A\to B$ and $g:B\to A$ be epimorphisms.
Are the rings $A$ and $B$ necessarily isomorphic?
[In this post "ring" means "commutative ring with one", and morphisms are required to map $1$ to $1$. By definition, a morphism of rings $f:A\to B$ is an epimorphism if for all pairs of morphisms $(g,h):B\rightrightarrows C$ the equality $g\circ f=h\circ f$ implies $g=h$. Surjective morphisms are epimorphic, but the converse does not always hold: for instance the inclusion $\mathbb Z\to\mathbb Q$ is an epimorphism.]
The busy reader is invited to skip the sequel.
For more details about epimorphisms see
$\bullet$ MathOverflow thread What do epimorphisms of (commutative) rings look like?.
$\bullet$ Stacks Project Section Epimorphisms of rings.
$\bullet$ Samuel Seminar . See in particular Section 2 of Exposé Number 7 by Daniel Ferrand.
The answers to the following variants of the above question are known:
(1) If $f:A\to B$ and $g:B\to A$ are injective morphisms of noetherian rings, are $A$ and $B$ necessarily isomorphic? The answer is No, as shown by the following example taken from a comment of Sam Lichtenstein to this question. Let $K$ be a field, $x$ an indeterminate, $f:K[x^2,x^3]\to K[x]$ the inclusion, and $g:K[x]\to K[x^2,x^3]$ the (clearly injective) morphism defined by $g(p(x))=p(x^2)$. Note that $K[x^2,x^3]$ is not isomorphic to $K[x]$ because the ideal $(x^2,x^3)$ of $K[x^2,x^3]$ is not principal.
(2) If $f:A\to B$ and $g:B\to A$ are surjective morphisms of rings, are $A$ and $B$ necessarily isomorphic? The answer is No, as shown by the following example taken from the same comment of Sam Lichtenstein. Set $$ A:=\mathbb Z/(4)\times\mathbb Z/(4)\times\cdots,\quad B:=\mathbb Z/(2)\times A, $$ let $f:A\to B$ be defined by $f(x_1,x_2,\dots)=(h(x_1),x_2,\dots)$, where $h$ is the unique ring morphism from $\mathbb Z/(4)$ to $\mathbb Z/(2)$, and let $g:B\to A$ be defined by $g(x_1,x_2,\dots)=(x_2,x_3,\dots)$. The rings $A$ and $B$ are not isomorphic because the equations $2x=0$ and $x^2=x$ have no nonzero simultaneous solutions in $A$, and one such solution in $B$ (namely $x=(1,0,\dots)$).
(3) If $f:A\to B$ and $g:B\to A$ are surjective morphisms of noetherian rings, are $A$ and $B$ isomorphic? The answer is Yes, because surjective endomorphisms of noetherian rings are isomorphisms. But epimorphic endomorphisms of noetherian rings are not always isomorphisms: see this answer of Eric Wofsey.