If $\mathbb{R}$ is the set of real numbers and $x \mathop{\#} y = \max\{x,y\}$ then $(\mathbb{R},\#,+)$ is a semiring where $(\mathbb{R},\#)$ is a semigroup and $+$ distributes over $\#$.
If you have a set $R$ with three distinct binary operations $*, +, \#$ such that $*$ distributes over $+$, and $+$ distributes over $\#$ then must $\mathop{\#}$ be an idempotent operation? (i.e. $x \mathop{\#} x=x$)
Does it make any difference if $(R,+,*)$ is a ring and $(R,\mathop{\#},+)$ is a semiring or does the double distributivity on its own force the idempotency?
Whether or not it turns out to be the case that # must be idempotent, does the double distributivity imply any other restrictions on the characteristics of $*$, $+$ or $\mathop{\#}$?
Just to get you started, try making $*$ something silly, say for example $R = \mathbb{R}$ and $x*y = 0$ for all $x,y\in R$. Then you should be able to choose + and # to be some standard operations such that all the required distributivity laws hold and # is not idempotent.
Of course in this case $(R,*,+)$ is not a ring. How might it help force # to be idempotent if it were a ring?