Unformatted text preview: pR prime ” p prime” is tautological. Finally, we prove the implication “ p prime H) p irreducible.” Sup-pose p is prime and p has a factorization p D ab . We have to show that either a or b is a unit. Because p is prime and divides ab , p divides a or b . Say p divides a , namely a D pr . Then p D ab D prb . Since R is an integral domain, we can cancel p , getting 1 D rb . Therefore, b is a unit. Now suppose that R is a principal ideal domain. To show the equiva-lence of the four conditions, we have only to establish the implication “ p irreducible H) pR maximal.” Suppose that p is irreducible and that J is an ideal with pR ´ J ´ R . Because R is a principal ideal domain, J D aR for some a 2 R . It follows that p D ar for some element r . Since p is irreducible, one of a and r is a unit. If a is a unit, then J D aR D R . If r is a unit, then a D r ± 1 p , so J D aR D pR . n...
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- Fall '08
- Algebra, Integral domain, Ring theory, Principal ideal domain, prime ideal, irreducibles