Suppose $M$ is a commutative cancellative monoid with $0$ as the identity and $+$ as the operation, and $M$ is equipped with an order $\preceq$ defined by $$ m \preceq m' \text{ if and only if there exists } x \in M \text{ such that } m + x = m'. $$ In order for $\preceq$ to be a partial order, no pair of non-zero elements $m, m'$ of $M$ can satisfy $m + m' = 0$. Assume this is the case. Then $0$ is the minimum of $M$.
My question is: If $R$ is an integral domain, will $R[M]$, the monoid ring of $M$, be an integral domain?
I tend to believe the answer is positive, but as I try to prove it, I am not sure anymore.