Let $(M,\times)$ be a monoid with zero. Let $\Sigma(M,\times)$ be the set of binary operations $+$ on $M$ such that $(M,+,\times)$ is a ring. Let $\sim$ be an equivalence relation on $\Sigma(M,\times)$ defined by $$+\_1\sim+\_2\iff(M,+\_1,\times)\cong(M,+\_2,\times).$$


Let's denote the quotient set $\Sigma(M,\times)/\sim$ by $\Sigma'(M,\times)$ and consider the number $$\mathrm{add}(M,\times)=|\Sigma'(M,\times)|.$$


I have several questions about the behavior of this function, none of which I know how to approach.


$(1)$ For an integer $n\geq 0$, is there always a monoid $(M,\times)$ such that $n=\mathrm{add}(M,\times)?$


$(2)$ The same question with the requirement that $M$ be finite.


$(3)$ The two previous questions are equivalent if $\mathrm{add}(M,\times)\geq\aleph\_0$ for $M$ infinite. Is it true? It is false by Todd Trimble's answer. $(\mathbb Z,+)$ with a zero element adjoined is another example. 


$(4)$ Is there an upper bound to the values of $\mathrm{add}(M,\times)$ over all finite monoids with $0?$ What about all monoids with zero?



For a monoid with zero $M$, how many additive operations on $M$ can there be making $M$ a ring?
I think the answer to (2) is yes (which also answers (1) and (4), of course).


First, note that $R=\mathbb{Z}/4\mathbb{Z}$ and $S=\mathbb{F}\_2[x]/(x^2)$ have isomorphic multiplicative monoids (via the map $0\mapsto 0$, $1\mapsto 1$, $2\mapsto x$, $3\mapsto 1+x$).


Call this monoid $M$.


Then the direct product $M^n$ of $n$ copies of $M$ is the multiplicative monoid of
$T\_k=R^k\times S^{n-k}$ for $k=0,\dots,n$, giving $n+1$ rings that are pairwise non-isomorphic, since they have different additive groups.


I claim that these are the only rings with $M^n$ as multiplicative monoid. Suppose $T$ is another. Then it follows just from the multiplicative structure that $T$ is commutative and has $n$ primitive idempotents $e\_1,\dots,e\_n$ such that, as a ring, $T\cong Te\_1\times\dots\times Te\_n$, where each $Te\_i$ is a ring whose multiplicative monoid is $M$.


But it's quite easy to see that $R$ and $S$ are the only rings with multiplicative monoid $M$: the subring generated by $1$ can only be $\mathbb{Z}/4\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z}$, so either the ring is isomorphic to $R$ or is a two-dimensional $\mathbb{F}\_2$-algebra generated by an element $x$ with square zero, in which case it must be isomorphic to $S$. 


