Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-isomorphism

$$\varphi:(R,\cdot)\cong(R,+)$$

and $\Bbb N$ is a sub-semiring of $R$. 

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**Observations**

The multiplication will be commutative. Also, there will be *many* new "numbers", e.g.

 - a unique additively absorbing element $\eta:=\varphi(0)$, i.e. $\eta+x=\eta$ for all $x\in R$.
 - many numbers with only a few additive decompositions as there are many natural numbers with only a few multiplicative decompositions: E.g. for prime numbers $p$, the number $\tilde p:=\varphi(p)$ cannot be written as a sum of two other numbers except $0+\tilde p$.

This is essentially as far as I came. No need to mention that I have not found a contradiction. Still hard to believe that it is possible.