Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes\_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume also that $k \longrightarrow A$ is injective. Then: Must $A$ be isomorphic to a $k$-subalgebra of $K$? If necessary, assume also that $k$ is an integral domain and/or is of characteristic zero.


Conversely: For which $k$-algebras $A$ contained in $K$ is the multiplication map $A \otimes\_k A \longrightarrow A$ an isomorphism? Equivalently, for which $k$-algebras $A$ contained in $K$ is $A \otimes\_k A$ $k$-torsion-free?



Tensor powers of an algebra all isomorphic
I think the missing word is *flat* : $A$ should be a flat finitely presented $k$-algebra. Indeed put $Y=\mathrm{Spec}\,k$ and $X=\mathrm{Spec}\,A$. The condition on $A/k$ is that the diagonal morphism $X\rightarrow X\times \_YX$ is an isomorphism. This means exactly that $f:X\rightarrow Y$ is unramified and universally injective ("radiciel"). If we add that $f$ is flat, then it is an open embedding (EGA IV, 17.9), which means $A\subset K$.


It is easy to give conter-examples if for instance we ask only for $k\rightarrow A$ to be injective.


