Name for a semiring minus multiplicative identity requirement Is there a name for a theory that has all axioms of a semiring but an axiom that mandates multiplicative identity?
Semiring theory is plagued by a mess of vocabulary the remains unresolved. Still, there are two books I always check when dealing with them.

In Golan's Semirings and Their Applications, a semiring without identity is called a hemiring.

A newer, also fairly comprehensive book on semirings is Graphs, Dioids and Semirings: New Models and Algorithms by Gondran and Minoux. They list semirings without identity as pre-semirings.

Hemiring in particular gets thousands of hits, although due to the mess of vocabulary in the theory, I can't guarantee what percentage of those is what you seek.

While I don't think those terms are going to be universally used, I think the references in both these books would help you track down specific information about such structures.
This would be the fusion of semirings and pseudo-rings (or rngs), so they should be called pseudo-semirings (or semirngs). But with google I couldn't find any usage for this (only with other meanings).