Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$.

I have two, probably trivial, questions:

Can one prove that $S$ satisfies the left Ore condition (for rings) as a multiplicative set in $\mathbb Z[S]$? I have tried both to find references and to prove it myself, but without success.

Even if the answer to (1) is negative, one can still consider the map $\mathbb Z[S]\to \mathbb Z[G]$ and use it to make a left $\mathbb Z[S]$-module $M$ into a left $\mathbb Z[G]$-module $$ \bar M:=\mathbb Z[G]\otimes_{\mathbb Z[S]}M. $$ Furthermore, there is a canonical map $\varphi \colon M\to \bar M$, defined by $\varphi(m):=1\otimes m$, for all $m\in M$. Is there an easy description for $\ker(\varphi)$? For example, is $\ker(\varphi)=\{m\in M:\exists s\in S, sm=0\}$?