About a year ago, a colleague asked me the following question:



> 
> Suppose $(R,+,\cdot)$ and $(S,\oplus,\odot)$ are two rings such that $(R,+)$ is isomorphic, as an abelian group, to $(S,\oplus)$, and $(R,\cdot)$ is isomorphic (as a semigroup/monoid) to $(S,\odot)$. Does it follow that $R$ and $S$ are isomorphic as rings?
> 
> 
> 


I gave him the following counterexample: take your favorite field $F$, and let $R=F[x]$ and $S=F[x,y]$, the rings of polynomials in one and two (commuting) variables. They are not isomorphic as rings, yet $(R,+)$ and $(S,+)$ are both isomorphic to the direct sum of countably many copies of $F$, and $(R-\{0\},\cdot)$ and $(S-\{0\},\cdot)$ are both isomorphic to the direct product of $F-\{0\}$ and a direct sum of $\aleph\_0|F|$ copies of the free monoid in one letter (and we can add a zero to both and maintain the isomorphism).


He mentioned this example in a colloquium yesterday, which got me to thinking: 



> 
> **Question.** Is there a counterexample with $R$ and $S$ finite?
> 
> 
> 



Non isomorphic finite rings with isomorphic additive and multiplicative structure
There do exist pairs of finite unital rings whose additive structures
are isomorphic and whose multiplicative structures are isomorphic,
yet the rings themselves are not isomorphic. 


To see this, let $\mathbb F$ be a field and let $X = \{x\_1,\ldots, x\_n\}$
be a set of variables. The polynomial ring $\mathbb F[X]$
is graded by degree
$$
\mathbb F[X] = H\_0\oplus H\_1\oplus H\_2\oplus\cdots.
$$
Let $Q(x\_1,\ldots,x\_n)$ be a quadratic form over $\mathbb F$.
Let
$$I = \mathbb F\cdot Q(X)\oplus H\_3\oplus H\_4\oplus\cdots$$
be the 
ideal generated by $Q(X)$ and the homogeneous components
of degree at least $3$.
Let $S\_{\mathbb F,Q}$ denote the $\mathbb F$-algebra
$\mathbb F[X]/I$. It is a commutative, local ring, which encodes
properties of the quadratic form $Q$.


Two quadratic forms $Q\_1$ and $Q\_2$ are equivalent
if they differ by an invertible linear change of variables.


**Claim.**
 Let $\mathbb F$ be a finite field of odd characteristic $p$.
Let $Q\_1(x\_1,\ldots,x\_n)$ and $Q\_2(x\_1,\ldots,x\_n)$ be 
nonzero quadratic forms over $\mathbb F$.


1. $S\_{\mathbb F,Q\_1}$ and $S\_{\mathbb F,Q\_2}$ have isomorphic
$\mathbb F$-space structures.
2. If $n>4$ and $Q\_1$ and $Q\_2$ are nondegenerate, then
$S\_{\mathbb F,Q\_1}$ and $S\_{\mathbb F,Q\_2}$ have
isomorphic multiplicative monoids.
3. $S\_{\mathbb F,Q\_1}\not\cong S\_{\mathbb F,Q\_2}$ as $\mathbb F$-algebras,
unless $Q\_1$ is equivalent to a nonzero scalar multiple of $Q\_2$.


*Proof.* Exercise! \\


So let $\mathbb F = \mathbb F\_3$ be the $3$-element field.
It is known that over a finite field of odd characteristic
the quadratic forms are classified by the dimension
and by the determinant of the form modulo squares.
The determinant of 
$$
Q(x\_1,\ldots,x\_n)=a\_1x\_1^2+a\_2x\_2^2+\cdots+a\_nx\_n^2
$$
is $a\_1\cdots a\_n$. If
$\alpha\in \mathbb F\_3^{\times}=\{\pm 1\}$,
then $\alpha\cdot Q$ has determinant
$\alpha^n a\_1\cdots a\_n=(\pm 1)^n a\_1\cdots a\_n$.
If $n$ is even, then the determinants of $Q$ and $\alpha\cdot Q$ will be equal,
so $Q$ will be equivalent 
to $\alpha\cdot Q$ for every $\alpha\in \mathbb F\_3^{\times}$.
This implies that, when working over $\mathbb F\_3$ in an even dimension,
if $Q\_1$ is not equivalent to $Q\_2$, $Q\_1$ will also
not be equivalent to any nonzero scalar
multiple of $Q\_2$.
In particular, no scalar multiple of
$$
Q\_1 = x\_1^2 + x\_2^2 + x\_3^2 + x\_4^2 + x\_5^2 + x\_6^2,
$$
is equivalent to 
$$
Q\_2 = x\_1^2 + x\_2^2 + x\_3^2 + x\_4^2 + x\_5^2 - x\_6^2
$$
over $\mathbb F\_3$. For these forms we have
that $S\_{\mathbb F,Q\_1}$ and $S\_{\mathbb F,Q\_2}$ are nonisomorphic
finite unital rings with isomorphic additive and multiplicative structures.
(These rings have size $3^{27}$.)


**Minor side comment 1:** If you allow nonunital rings, there is a pair of nonisomorphic $8$-element rings whose additive and multiplicative structures are isomorphic.


**Minor side comment 2:**
The solution to the exercise above (that is, the proof of the Claim)
can be found [here](http://spot.colorado.edu/~kearnes/MathOverflow/MO-8-22-15.pdf).


