3330_Notes_5o4_fill - x y if and only if x y D-∈...

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Math 3330 Section 5.4 Ordered Integral Domains The “short” version- Define an order on an integral domain just like we defined the set of positive integers in chapter 2. An integral domain D is called an ordered integral domain if D contains a subset D + Such that D + has the following properties: 1. D + is closed under addition 2. D + is closed under multiplication 3. for every x in D, exactly one of the following is true: x D + , x 0 = or x D + - ∈ Define an order “>” by
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Unformatted text preview: x y if and only if x y D +-∈ Properties – similar to integers 1( the unity) is positive The square of an element is positive If D is an integral domain such that D + is well-ordered (every nonempty subset has a least element), then D is ring isomorphic to the ring of integers. So Z is “the well-ordered” integral domain. You can also define an ordered field in the same way. Then R is an ordered field....
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