section_path: Order-theoretic properties
section_name: Order-theoretic properties
paragraph_text: Order-theoretic properties
  
  Z {\displaystyle \mathbb {Z} } is a totally ordered set without upper or lower bound. The ordering of Z {\displaystyle \mathbb {Z} } is given by::... −3 < −2 < −1 < 0 < 1 < 2 < 3 <... An integer is positive if it is greater than 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:
  
  Thus it follows that Z {\displaystyle \mathbb {Z} } 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. This is equivalent to the statement that any Noetherian valuation ring is either a field —or a discrete valuation ring.