# Name for semiring like structure

Is there a name for the structure $(G,+,\cdot)$ where

1. $(G,+)$ is a commutative monoid and
2. 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.

Thanks.

• @BillTrok Near-semirings only have one distribution law and also one absorption rule ($0\cdot a = 0$) – Stefan Perko Jun 5 '15 at 20:02
• @StefanPerko whoops you are correct – Bill Trok Jun 5 '15 at 20:07
• This book bestows many names upon many combinations of subsets of ring axioms. It may or may not appear in that book. Even if it does, the term used there may not be in common use. – rschwieb Jun 5 '15 at 20:37

A rig is a ring without additive inverses (it's missing negatives). So $(G,+)$ in this case is only required to be a commutative monoid. A rg (spoken as rug) is a rig without multiplicative identity (it's missing an identity). Now, your structure has to be a nonassociative rg.