Prove that if D is a wellordered integral domain, a ED, a =I 0, and a =I e, then a2 > a

Step 1 of 4

Definition-1: An integral domain D is said to be ordered if there is a subset of D such that:

(i) Closure under addition: If , then

(ii) Closure under multiplication: If , then

(iii) Law of trichotomy: If , then exactly one of the following is true, or

Note: The elements of are called the positive elements of D

Definition-2: Assume that D is an ordered integral domain and . Then will mean that  . If , we say that a is greater than b and that b is less than a

Lemma-1: If and, then

Lemma-2: If D is well ordered integral domain with unity e, then e is the least element of

Corollary-1: If D is an ordered integral domain and , then