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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).

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.