Does there exist a non-trivial commutative binary operation on $\Bbb C$ that distributes over both multiplication and addition?
In other words, if our operation is denoted by $\odot$, then I want the following to hold:
- $a \odot (b \cdot c) = a \odot b \cdot a \odot c$
- $a \odot (b + c) = a \odot b + a \odot c$
- $a \odot b = b \odot a$
All of the things I can find so far distribute over either multiplication or addition, but not both. Alternatively, is there a proof that no such operation can exist?
I wasn't sure if this question was too elementary for MO, so I'm trying here first. This question is obliquely related to the following other questions I've asked on MO and here: