(adsbygoogle = window.adsbygoogle || []).push({}); Fact:The ring of integersZis totally ordered: for any distinct elementsaandbinZ, eithera>bora<b.

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

Observation:InZ, 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 asZ, assuming there are any other?

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# Multiplicative and additive identities as successors

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