Is there an algebraic structure over an infinite set $S$, composed by $3$ different binary operators $*$, $+$,$×$ such that :
- $*$ form an abelian group over $S$ and let's call $e$ its identity element
- $*$ and $+$ form a field over $S$
- $+$ and $×$ form a field over $S\setminus\{e\}$
If this structure exist what is its name? Can you give an example? If it does not exist, why? What is "wrong" in this structure?