Comment by areyousure
4 days ago
> Ask four mathematicians whether something is a Commutative Ring or not and they'll all agree.
Amusingly, there are at least 3 different definitions of commutative ring.
The principal issue is whether it must have a 1 (unity, ie a multiplicative inverse). Wikipedia https://en.wikipedia.org/wiki/Commutative_ring as well as most modern sources insist on this.
Britannica https://www.britannica.com/science/ring-mathematics#ref89421... as well as many older sources (such as Noether's original definition and van der Waerden) do not insist that the ring have a 1. Even first-edition Bourbaki didn't have 1!
Finally, if you do have a 1, then sometimes people include the condition that 0 != 1, ie the trivial/zero ring is deemed not a [commutative] ring. This is somewhat hard to find, but is relatively common among people who specifically define the concept of "ring with identity" (eg Zariski+Samuel). I have also found it unqualified (ie, just in the definition of "commutative ring") in the wild, eg in "Handbook of Mathematical Logic" by Barwise or "The Math You Need" by Mack.
(I agree with people like Conrad and Poonen that rings should have a 1. And I guess that the zero ring is in fact a [commutative] ring.)
The moral of the story is: "be careful with simile and analogy". Otherwise you get your arse handed to you on a plate and your very reasonable argument gets lost in the weeds 8)
In these circles is is generally safer to stick with car analogies.