Suppose $S$ is an associative semiring whose underling commutative monoid is free (in particular, cancellative) and that its Grothendieck ring $G(S)$ is a unital ring. Can we conclude that $S$ must be unital, and if not, is there a nice counter-example?

Alternatively, are there additional suppositions we can put on $S$ which would allow us to conclude the unitality of $S$ from that of $G(S)$?

The context is that I am struggling to disprove that a particular ring I have constructed is unital, but I have proven it is of the form $G(S)$ for $S$ which is not unital, and I am hoping this helps me somehow.



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