Let $M$ be a non-trivial monoid and $\mathbb ZM$ its monoid ring. All modules are left modules in what follows. Suppose that $M$ contains a zero element (or absorbing element) $z$. That is $mz=z=zm$ for all $m\in M$. For example the zero element of a ring is absorbing for multiplication. It is well known that the trivial $\mathbb ZM$-module $\mathbb Z$ is projective in this case. Indeed, $z$ is a central idempotent and $\mathbb ZMz\cong \mathbb Z$. Therefore, $\mathbb ZM=\mathbb ZMz\oplus \mathbb ZM(1-z)$ and hence $\mathbb Z$ is projective. Moreover, the above direct sum is a ring direct product decomposition.

Question 1.Can the trivial module $\mathbb Z$ be stably free?

Recall that a finitely generated projective module $P$ is *stably free* if $P\oplus F\cong F'$ for some finitely generated free modules $F,F'$. Note that in our setting if $\mathbb Z\oplus F\cong F'$ and $r$ is the rank of $F$ and $r'$ is the rank of $F'$, then

$$\mathbb Z^{r'}\cong zF'\cong \mathbb Z\oplus zF\cong \mathbb Z^{r+1}$$ and so $r'=r+1$ if this happens.

Also, $R=\mathbb ZM(1-z)$ is a ring, known as the *contracted monoid ring* of $M$. It is obtained by identifying the zero of $M$ with the zero of $\mathbb ZM$ and it has the property that $R$-modules are precisely those $M$-modules annihilated by the zero of $M$.

Then note that

$$R^{r+1}\cong (1-z)F'\cong (1-z)F\cong R^r$$

and hence if $\mathbb Z$ is stably free, then $R$ does not have the *Invariant Basis Number Property*, which says that finite rank free modules "know" their rank.

Question 1'.Can a contracted monoid ring fail to have the Invariant Basis Number Property?

If either $M$ is finite or $M\setminus \{z\}$ is a submonoid, then the contracted monoid ring has the Invariant Basis Number Property.

It has been suggested to me by both George Bergman and Jason Bell (independent private communications) that the polycyclic inverse monoid (or Cuntz monoid)

$$P_2=\langle x,x^*,y,y^*,z\mid xx^*=1,yy^*=1, xy^*=z=yx^*, z=0\rangle,$$

where $z=0$ is shorthand for $z$ is the absorbing element, may be relevant here for Question 1'.

The point is that if $R$ is the contracted monoid ring of $P_2$, then $R^2$ is isomorphic to a direct summand in $R$. One notes that $x^*x,y^*y$ are orthogonal idempotents and so $e=x^*x+y^*y$ is an idempotent and $(a,b)\mapsto ax+by$ is a module isomorphism $R^2\to Re$ with inverse $c\mapsto (cx^*,cy^*)$. Thus $Re$ is stably isomorphic to $R$. Since $e\neq 1$, this doesn't contradict the invariant basis number property, but it hints that the $K$-theory of this ring may not be so nice.

Note that $R/R(1-e)R=R/R(1-(x^*x+y^*y))R$ is a Leavitt algebra which fails to have the invariant basis number property because the free modules of rank $1$ and $2$ are isomorphic. The extension of scalars of this algebra over any field is simple and this ring is known to contain interesting groups inside of its group of units like the simple group Thompson's group $V$.