This question is somewhat related to Tilmans notorious problem in [this post][1]. Let $(M,\cdot)$ be a monoid with unit $1$ and set
$$(M,\cdot)^{\times} := \lbrace x \in M \mid \exists y \in M : xy=yx=1 \rbrace.$$
Let $k$ be a field (say $\mathbb Z/ 2 \mathbb Z$) and let $k[M]$ be the monoid ring of $M$ with coefficients in $k$. Consider now $$GL(k[M]) := (k[M],\cdot)^{\times}.$$

> **Question:** Can it happen that $(M,\cdot)^{\times} = \lbrace 1 \rbrace$ but $\lbrace 1 \rbrace \subsetneq GL(k[M])$?

Clearly, if $\lbrace 1 \rbrace \subsetneq GL(k[M])$, then $M$ must contain right- and left-invertible elements; but that is not enough to ensure the existence of invertible elements.

  [1]: http://mathoverflow.net/questions/17532/does-linearization-of-categories-reflect-isomorphism