Editing Integer (section)

==Order-theoretic properties==
<math>\mathbb{Z}</math> is a [[total order|totally ordered set]] without [[upper and lower bounds|upper or lower bound]]. The ordering of <math>\mathbb{Z}</math> is given by:
{{math|:... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ...}}.
An integer is ''positive'' if it is greater than [[0|zero]], and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:
# If {{math|''a'' < ''b''}} and {{math|''c'' < ''d''}}, then {{math|''a'' + ''c'' < ''b'' + ''d''}}
# If {{math|''a'' < ''b''}} and {{math|0 < ''c''}}, then {{math|''ac'' < ''bc''}}

Thus it follows that <math>\mathbb{Z}</math> together with the above ordering is an [[ordered ring]].

The integers are the only nontrivial [[totally ordered]] [[abelian group]] whose positive elements are [[well-ordered]].<ref>{{cite book |title=Modern Algebra |series=Dover Books on Mathematics |first=Seth |last=Warner |publisher=Courier Corporation |year=2012 |isbn=978-0-486-13709-4 |at=Theorem 20.14, p.&nbsp;185 |url=https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185 |access-date=2015-04-29 |archive-url=https://web.archive.org/web/20150906083836/https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185|archive-date=2015-09-06 |url-status=live}}.</ref> This is equivalent to the statement that any [[Noetherian ring|Noetherian]] [[valuation ring]] is either a [[Field (mathematics)|field]]—or a [[discrete valuation ring]].