Ringoid

A ringoid $R$ is a set $(R,+,\times)$ with two binary operators, conventionally denoted addition ($+$) and multiplication ($\times$), where $\times$ distributes over $+$ left and right:

$$a(b+c)=ab+ac\quad\hbox{and}\quad(b+c)a=ba+ca.$$

A ringoid can be empty.

See also Binary Operator, Ring, Semiring

References

Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968.