The axioms of a semiring include:
$$0·a = 0 = a·0$$
Is there a name for an algebraic structure which satisfies all the axioms of a semiring except for this one?
(if it helps, the structure I have in mind happens to satisfy $0·a·0=0$)
If you were looking for a field with convenient terminology for its structures, I should warn you that the field of semirings is pretty bad :)
"Graphs, dioids and semirings: new models and algorithms" by Michel Gondran, Michel Minoux
they use "presemiring" to mean a set with two associative binary opearations, + and X where + is commutative and X distributes over + on both sides.
To make a presemiring a semiring, they require identity elements for both operations, and require that the additive identity 0 is absorbing, as you describe.
I wouldn't put too much stock in the one book though. Really there are so many authors throwing around so many terms about these things it's probably useless to find out which is the most common.