# Show that if a submonoid $S$ of a commutative monoid $M$ contains an absorbing element, then the localization of $M$ by $S$ contains only one element.

Here is the full question:

If $$M$$ is a commutative monoid, $$S\subset M$$ is a submonoid and there is a $$z\in S$$ that is absorbing in $$M$$ (i.e. $$zm=z$$ for all $$m\in M$$), then show that $$M_S$$ has only one element.

I know that this is a divide-by-zero type of question, and I'm just not sure how to word this. Here's what I've written so far.

We want to construct a monoid of fractions, $$M_S=S^{-1}M$$, where we can write the congruence classes of $$M_S$$ as $$\frac{a}{s}=[(a,s)]$$. From Theorem 1, part (2) of the handout, we need that for every $$s \in S$$, then $$\phi(s)=[(s,e)]$$ has an inverse $$[(e,s])$$ in $$M_S$$.

For $$z \in S$$, consider $$\phi(z)$$. Theorem says that $$\phi(z)=[(z,e)]$$ must have an inverse, $$[(e,z)]$$. So then we must have $$[(z,e)][(e,z)]=[(e,e)]$$. But $$z$$ is absorbing, so then $$[(z,e)][(e,z)]=[(ze,ez)]=[(z,z)]$$, and we have $$[(e,e)]=[(z,z)]$$.

I'm not sure how to proceed from here, or if what I wrote above is on the right track.

Additionally, here is the Theorem that I cite (which we are allowed to do):

Let $$M$$ be a commutative monoid, $$S$$ be a submonoid all of whose elements are cancellable in $$M$$ and $$M_S$$ be the localization of $$M$$ by $$S$$.

1. The function $$\phi:M\rightarrow M_S$$ given by $$\phi(m)=[(m,e)]$$ is an injective monoid homomorphism.
2. For every $$s\in S$$, $$\phi(s)=[(s,e)]$$ has inverse $$[(e,s)]\in M_S$$.
3. If $$\psi:M\rightarrow N$$ is a homomorphism of commutative monoids such that $$\psi(s)$$ is invertible in $$N$$ for every $$s\in S$$, then there is a unique monoid homomorphism $$\rho:M_S\rightarrow N$$ such that $$\psi=\rho\phi$$.

$$\rho$$ is given by the formula $$\rho([(m,s)])=\psi(m)*\psi(s)^{-1}$$.

Thanks ahead of time for any feedback.

• Doesn’t this follow directly from the definition of equivalence for $M_S$? If I recall correctly, it’s similar to the one for ring localization... Namely, isn’t $[(m,s)] = [(n,s’)]$ if and only if there exists $t\in S$ such that $tms’ = tns$? – Arturo Magidin Apr 6 at 21:07