It is well known that a localization $S^{-1}R$ of a commutative ring $R$ is flat as a $R$-module.

Rather, I am looking for extensions of rings which share certain properties of localizations, like flatness, while inverting nothing.

More precisely, I would like to find classes of examples, if such exist, of maps of monoids $f:M\to N$ verifying

$N^{\times} = \{1\}$,

$f$ is injective, but not surjective,

"$M$ is cofinal in $N$" in the sense that the augmentation induces an isomorphism $\Bbb{Z}[N]\otimes_{\Bbb{Z}[M]}\Bbb{Z}[1]\cong\Bbb{Z}[1]$ of abelian groups (note that $\Bbb{Z}[1]\cong\Bbb{Z}$ as abelian groups),

for which $\Bbb{Z}[N]$ is flat as either a left or right $\Bbb{Z}[M]$-module.

The first two conditions listed above are simply to guarantee that $f$ does not invert any elements whatsoever, while not being an isomorphism.

For completeness, and in case it is easier, I am actually looking for examples of maps $f:M\to N$ for which $\text{Tor}^{\Bbb{Z}[M]}_\ast(\Bbb{Z}[N],\Bbb{Z}[N])$ is $\Bbb{Z}[N]$ concentrated in degree zero.

Remark: this condition is related to the forgetful functor from the positive derived category of $\Bbb{Z}[N]$-modules to the derived category of $\Bbb{Z}[M]$-modules being full and faithful.

Edit: I have removed the extraneous commutativity condition on the monoids.