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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

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

2. $f$ is injective, but not surjective,

3. "$M$ 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 two first 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.

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

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

2. $f$ is injective, but not surjective,

3. "$M$ 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.

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 commutative monoids $f:M\to N$ such that

1. $N^{\times} = \{1\}$;

2. $f$ is injective, but not surjective;

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

for which $\Bbb{Z}[N]$ is a flat module over $\Bbb{Z}[M]$.

The two first conditions listed above are simply to guarantee that $f$ does not invert any elements whatsoever, while not being an isomorphism. Furthermore, I have assumed commutativity of the monoids for convenience.

For completeness, and in case it is easier, I am actually looking for examples in 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.

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

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

2. $f$ is injective, but not surjective,

3. "$M$ 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 two first 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.

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

  I am looking for examples which are essentially orthogonal to these. I would like to find classes of examples, if such exist, of maps of commutative monoids $f:M\to N$ such that

1. $N^{\times} = \{1\}$;

2. $f$ is injective, but not surjective;

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

for which $\Bbb{Z}[N]$ is a flat module over $\Bbb{Z}[M]$.

The two first conditions listed above are simply to guarantee that $f$ does not invert any elements whatsoever, while not being an isomorphism. Furthermore, I have assumed commutativity of the monoids for convenience.

For completeness, and in case it is easier, I am actually looking for examples in 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.

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 commutative monoids $f:M\to N$ such that

1. $N^{\times} = \{1\}$;

2. $f$ is injective, but not surjective;

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

for which $\Bbb{Z}[N]$ is a flat module over $\Bbb{Z}[M]$.

The two first conditions listed above are simply to guarantee that $f$ does not invert any elements whatsoever, while not being an isomorphism. Furthermore, I have assumed commutativity of the monoids for convenience.

For completeness, and in case it is easier, I am actually looking for examples in 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.