A set with two binary operations?

Question

Hyello guys!

So, I'm wondering about whether there is a name for a set $S$ with two binary operations $+$ and $\times$ such that

$(S,+)$ is a commutative monoid, and

$\times$ is associative and has identity (with respect $\times$; so $(S,\times)$ is a monoid) and

$\times$ is distributive over $+$...

Thank you!

Answer 1

If you add in the condition that the zero element acts as an annihilator on the entire set, this is referred to as a semiring.