Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)\cong\text{Hom}(M,\text{Hom}(N,L)).$$ If $M$ and $N$ are abelian groups, then $M\otimes N$ agrees with the abelian group tensor product.
Now I would like to understand the derived tensor product and $\text{Tor}(M,N)$. Similar questions have been asked before, but I am interested in a very particular construction:
If we identify $M$, $N$, $\mathbb{N}$ as discrete commutative monoid spaces, we may take the relative smash product of $E_\infty$-spaces (see below) $M\wedge_{\mathbb{N}} N$. Each point $x\in M\otimes N$ corresponds to a connected component of $M\wedge_{\mathbb{N}} N$, and so we can take Tor groups based at x: $$\text{Tor}_n(M,N;x)=\pi_n(M\wedge_{\mathbb{N}} N;x).$$ If $M$ and $N$ are groups, the choice of basepoint is irrelevant (we can always shift to $x=0$), and we recover the usual Tor groups.
How might I go about computing this commutative monoid Tor? Here is a more specific question: if $\mathbb{N}\rightarrow R$ is a surjection of commutative semirings (such as $R=\mathbb{N}/(n=n+1)$), is there some $x\in R\otimes R\cong R$ such that $\text{Tor}(R,R;x)$ is nonzero? I suspect the answer is yes when $x=n$.
By the smash product of $E_\infty$-spaces, I mean for example $\infty$-categorical Day convolution of $\Gamma$-spaces, which satisfies the same tensor-Hom adjunction universal property. More generally, I would be interested in techniques to compute smash products of $E_\infty$-spaces, but I think this is very difficult, so I am asking about this toy example.
