$C_2$ denotes the cyclic group of order 2. How to find all ring structures over $C_2\times C_2$?
The question is equivalent to give a full list of all essentially different bilinear 2-operations over the abelian group $C_2\times C_2$ that is associative. I don't require those ring structures being unital, and two operations $\cdot$ and $\circ$ are said to be essentially different if there is no additive automorphism $f:C_2\times C_2\to C_2\times C_2$ such that $\forall x,y\in C_2\times C_2,\ x\cdot y=f(x)\circ f(y)$.
By the way, I've found that there are exactly three unital ring structures: $\mathbb F_4,\ \mathbb F_2\times\mathbb F_2$ and $\mathbb F_2\left[x\right]/\left(x^2\right)$.
It will help a lot if someone can tell me if there is any systematic method to determine such a list.
Hint: For singly generated ring structures $C_2\times C_2$, express them as a (non-unital) subrings of quotients of $\mathbb F_2[X]$. And for non-singly generated ring, express them as (not necessarily unital) subrings of $M_2(\mathbb F_2)$.