Is there a name for the structure $(G,+,\cdot)$ where
- $(G,+)$ is a commutative monoid and
- left and right distributive laws hold?
($(G,\cdot)$ is not necessarily associative or commutative and need not have an identity.) It seems that such a structure may have an established name but I cannot find anything online or in my algebra books. It is not a semiring, nonassociative ring, or any other named structure that I can find. It would be great if there is a reference that I can look up and cite.