Let $K_1,\dotsc,K_n$ be finite fields and let $V$ be the variety of rings, generated by the $K_i$ (rings aren't necessarily unital). I want to figure out what $V$ looks like. By a theorem of Tarski, elements of $V$ are the quotients of subrings of direct products of (possibly infinite) families of the $K_i$. But what are these rings exactly?
One thing we can figure out are the possible characteristics of the elements of $V$. Since taking subrings and quotients decreases the characteristic and the characteristic of the product is the least common multiple of the characteristics of the factors, any ring in $V$ has a characteristic which is a squarefree integer, whose prime factors are among the characteristics of the $K_i$. On the other hand, not every such ring lives in $V$. For example, the multiplicative semigroup of any ring in $V$ must have finite exponent (since this is true for the products of the $K_i$), which means that things like polynomial rings over $K_i$ can't appear in $V$.
I tried looking at the simplest case where $V$ is generated by $\mathbb{Z}_2$, but I can't really picture what's going on. I have a feeling that in this case $V$ will be the class of Boolean rings, but I'm not even sure how to show this.