Terminology for certain monoids which are to monoids like fields are to rings 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).
 A: The usual term in semigroup theory for a group with adjoined zero is a group with zero. See The Algebraic Theory of Semigroups Volume I by Clifford and Preston. 
A: One should take seriously the option of simply calling them "abelian groups with an adjoined zero".
A: One possible alternative name for monoids with zero would be $\mathbb{F}_{1}$-algebras: just like $R$-algebras are monoids in $(\mathsf{Mod}_{R},\otimes_{R},R)$,  monoids with zero are monoids in $(\mathsf{Mod}_{\mathbb{F}_{1}},\otimes_{\mathbb{F}_{1}},\mathbb{F}_{1})\mathbin{``="}(\mathsf{Sets}_*,\wedge,S^0)$.
Now, every object in $(\mathsf{Sets}_*,\wedge,S^0)$ comes with a natural diagonal morphism $X\to X\wedge X$ given by the composition $X\to X\times X\twoheadrightarrow X\times X/X\vee X$ and also a projection map $X\twoheadrightarrow S^0$.
Because of these maps, every monoid with zero $M$ satisfying $M^\times=M\setminus\{0\}$ will be an example of a "Hopf $\mathbb{F}_{1}$-algebra", a Hopf algebra object in $(\mathsf{Sets}_*,\wedge,S^0)$.
Not every such Hopf $\mathbb{F}_{1}$-algebra arises as such an $M$, but this seems like a notion that is quite close to what you are looking for.
Every Hopf $\mathbb{F}_{1}$-algebra arises in this way because the categories of comonoids in $(\mathsf{Sets},\times,*)$ and $(\mathsf{Sets}_*,\wedge,S^0)$ are equivalent, see Lemma 2.4 here. So your notion is equivalent to that of a Hopf $\mathbb{F}_{1}$-algebra, i.e. a Hopf algebra object in $(\mathsf{Sets}_*,\wedge,S^0)$.
