Let $k$ be a field. Let $A$ and $B$ be two $k$-algebras, ie. two rings that are also $k$-vector spaces and their multiplication is $k$-bilinear.
Any isomorphism of $k$-algebras is also a ring isomorphism, so if $A$ and $B$ are isomorphic as $k$-algebras, they are isomorphic as rings.
I would guess that the converse fails. Is there any example of $A$ and $B$ that are isomorphic as rings, but not as $k$-algebras?
The reason I came up with this question is the following. Two affine varieties are isomorphic if and only if their coordinate rings are isomorphic as $k$-algebras. I am interested in finding an example where coordinate rings are isomorphic as rings, but the varieties are not isomorphic.