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.
A monoid with an distinguished absorbing element is called a monoid with zero in the literature. A monoid with zero in which the units are exactly the nonzero elements is called a group with zero [1, p. 5; 2, Def 1.3.1, p. 34; 4] or a $0$-group [3].
[1] A. H. Clifford, G. B. Preston, The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7 American Mathematical Society, Providence, R.I. 1961 xv+224 pp.
[2] P.M. Higgins, Techniques of semigroup theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+258 pp. ISBN: 0-19-853577-5
[3] J.M. Howie, Fundamentals of semigroup theory. London Mathematical Society Monographs. New Series, 12. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+351 pp. ISBN: 0-19-851194-9
[4] Lallement, Gérard. Semigroups and combinatorial applications. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, New York-Chichester-Brisbane, 1979. xi+376 pp. ISBN: 0-471-04379-6