TITLE: Is there a name for a monoid with a distinguished absorbing element?
QUESTION [6 upvotes]: Let $M = (M,·,1,0)$ be a monoid $(M,·,1)$ together with an distinguished absorbing element $0 ∈ M$, that is such that $∀x ∈ M\colon 0·x = 0 = x·0$. 
Does such a structure $M$ have a nice name?
Furthermore, is there a name for such structures $M$, where the units of $M$ are exactly the nonzero elements of $M$, i.e. $M^× = M\setminus \{0\}$?
Example. Every ring is such a structure when only considering multiplication. Fields are then examples where the units are exactly the nonzero elements.

REPLY [2 votes]: They are sometimes also called pointed monoids. See arXiv:1909.00297 for instance.