Ordered Ring

• An ordered ring is a ring with a nonempty subset (called the set of positive elements) satisfying two properties

  • Closure: ,
  • Trichotomy: one and only one of the following holds:

• Rules for multiplying positives and negatives hold:

  • If and where , then .
  • If where , then

• (Fraleigh e25.26) If is an ordered ring with as positive elements and , then satisfies the requirements for a set of positive elements in the ring . gives an ordering of .

• (Fraleigh 25.3) Let be an ordered ring, all squares of nonzero elements in are positive. and there are no zero divisors

  • Proof: Trichotomy gives or . Closure gives and
  • Thus, no finite ring can be ordered since the characteristic of an ordered ring is .

• (Fraleigh 25.4) can be considered as embedded in any ordered ring . The induced ordering from from is the natural ordering of (i.e. is the set of positive integers).

• (Fraleigh 25.5) Let be an ordered ring with set of positive elements. Let (is less than) be a relation defined by

  For . Then has the following properties

  • Trichotomy: One and only one of the following holds:
  • Transitivity:
  • Isotonicity:

  Conversely, given a relation on a nonzero ring satisfying the above, the set satisfies the criteria for a set of positive elements.

  • and in fact .
  • (Fraleigh e25.18)
  • (Fraleigh e25.19)
  • (Fraleigh e25.20)
  • (Fraleigh e25.21)

• An ordering of a ring with the property that , there exists a positive integer such that

  is called an Archimedean ordering

• (Fraleigh 25.10) Let be an ordered ring with set of positive elements and let be a ring isomorphism. The subset gives a set of positive elements of .

  Also, the ordering relation given by (denoted ) gives us

  That is, isomorphism between ordered rings preserves the ordering.

  The ordering is called the ordering induced by

Link §

• Fraleigh