definition of linearly-ordered ring
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
- The reader knows a definition of linearly-ordered set.
Target Context
- The reader will have a definition of linearly-ordered ring.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*R\): \(\in \{\text{ the rings }\}\), with any linear ordering, \(\lt\), that satisfies the conditions specified below
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Conditions:
\(\forall r_1, r_2, r_3, r_4 \in R \text{ such that } r_1 \le r_3 \land r_2 \le r_4 (r_1 + r_2 \le r_3 + r_4)\)
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2: Note
\(r_1 r_2 \le r_3 r_4\) does not need to hold: for example, \(R = \mathbb{Z}\) with the canonical ordering is a linearly-ordered ring, but while \(- 2 \le 1\) and \(- 4 \le 3\), \((1) (3) = 3 \lt 8 = (-2) (-4)\).
This concept is not 'any ring, which happens to have any linear ordering' but 'any ring with any linear ordering whose (the ring's) addition preserves the ordering'.
Any linearly-ordered ring is a partially-ordered ring.
