Rings and idempotent semirings

Question

If $\mathbb{R}$ is the real numbers and x#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 # be an idempotent operation? (i.e. x#x=x)

Does it make any difference if (R,+,*) is a ring and (R,#,+) 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 #?

Answer 1

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?