Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "Schemes over $\mathbb{F}_1$ and Zeta functions" by Connes and Consani. However they don't give these monoids a name.

A very silly idea might be to call them "monoid fields".

**Question.** How are these monoids called in the literature? If there is no existing terminology yet, which one would you propose?

The answer by BS tells us that in the non-commutative case these are called *groups with zero*. My question deals with the commutative case. I would like to have a proper name, not just a combination such as "abelian group with zero" (which is confusing anyway).