What is a semiring without commutativity requirement for addition? A semiring is defined by two operations $+$ and $\times$ on a set $R$, such that:


*

*$(R, +)$ is a commutative monoid

*$(R, \times)$ is a monoid

*$\times$ left-distributes and right-distributes over $+$

*The identity element for $+$ is an absorbing element for $\times$


What is the name of a structure where $(R, +)$ is only required to be a monoid? I'm interested in this because it may make sense in the study of provenance semirings in relational databases. However, beyond this specific context, I think this is a natural subset of the semiring axioms.
The same question has been asked here for rings (where $(R, +)$ is required to be a commutative group), here and here. In this context, the commutativity requirement can be explained (as in this answer) because commutativity of $+$ is implied by distributivity, the existence of a neutral element for $\times$, associativity of $+$, and cancellativity of $+$. However, in a semiring, $(R, +)$ is only a monoid, so in general it is not cancellative: hence, the same explanation does not work.
A generalization of semirings that removes the commutativity requirement are near-semirings, but they additionally remove the requirement of a neutral element for $\times$ and one of the distributivity requirements (e.g., right-distributivity). Hence, my question is equivalent to asking whether there is a name for near-semirings with multiplicative identity and left- and right-distributivity of $\times$ over $+$.
Another related question here asks about just removing the identity requirement for $\times$. I checked quickly the references from this answer to see whether my concept had a name, but apparently Golan's Semirings and Their Applications and Gondran and Minoux's Graphs, Dioids and Semirings: New Models and Algorithms both assume commutativity of $+$ straight away.
 A: (Adding an answer to document what I have so far, also to avoid auto-cleanup of the question.)
The notion of "additively non-commutative semirings" is mentioned p30 of
Hebisch and Weinert, Semirings: Algebraic Theory and Applications in
Computer Science (or p33 of the German version Halbringe: Aigebraische Theorie und Anwendungen in der Informatik. They point to several papers:


*

*M. P. Grillet. Embedding of a semiring into a semiring with identity.
Acta Math. Acad. Sci. Hungar., 20:121 - 128, 1969.

*H. J. Weinert und R. D. Griepentrog. Embedding semirings by
translational hulls. Semigroup Forum, 14:235 - 246, 1977.

*R. D. Griepentrog und H. J. Weinert. Embedding semi rings in semi-
rings with identity. Coll. Math. Soc. Janos Bolyai, 20. Algebraic
Theory of Semigroups, North-Holland, S. 225 - 245, 1979.

*R. D. Griepentrog und H. J. Weinert. Correction and remarks to
our paper "Embedding semirings in semirings with identity". Coll.
Math. Soc. Janos Bolyai, 99. Semigroup,~, North-Holland, S. 491-493, 1985.


I was unable to find the last two online. The 1977 paper by Weiner and Griepentrog uses "semiring" to mean a "left and right distributive seminearring", where what they call a seminearring is almost like Wikipedia's near-semirings, but addition is a semigroup rather than a monoid (so the axiom about absorptivity of 0 is also not imposed). So their "semirings" are like what I asked about, except that they do not require the existence of a additive identity nor of a multiplicative identity (and hence not the absorptivity axiom). They do not seem to have a name to distinguish their concept from usual semirings. The paper by Grillet seems to be implicitly defining "semiring" to something similar to that.
It seems that the bottom line is that the concept I propose (or further weakenings thereof) was indiscriminately called "semiring" in the literature, but there is no dedicated term, and the terminology of these kinds of objects seems quite fuzzy.
