Multiplicative and additive identities as successors

[pilpel]

Fact: The ring of integers Z is totally ordered: for any distinct elements a and b in Z, either a>b or a<b.

Fact: The ring of integers is discrete, in the sense that for any element a in Z, there exists an element b in Z such that there is no element c in Z with a<c<b, and the same argument holds with the greater than signs flipped. In other words, successors and predecessors exist in Z (but not in R, for example).

Observation: In Z, the multiplicative identity 1 is the successor of the additive identity 0.

Questions: Is this fact a coincidence? Does this have any significance? Must the multiplicative identity always succeed the additive identity in rings that have the same properties as Z, assuming there are any other?

 

[fresh_42]

It has a significance. One is the smallest countable unit and as such the multiplicative unit as well as the smallest entity to count, hence the successor of zero.

If a ring has the same properties as $\mathbb{Z}$ then it is $\mathbb{Z}$.