MA3412S2_Hil2014.pdf

# Conversely let x be an element of r for which the

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Conversely let x be an element of R for which the corresponding principal ideal ( x ) is a non-zero prime ideal of R . Then x 6 = 0 R . Also ( x ) is a proper ideal of R , because the definition of prime ideals requires such ideals to be proper ideals of R , and therefore x is not a unit of R . Let y and z be elements of R . Then yz ( x ) if and only if either y ( x ) or z ( x ). It follows that x | yz if and only if either x | y or x | z . Therefore x is a prime element of R . Definition Let I 1 , I 2 , . . . , I k be ideals of a unital commutative ring R . The product I 1 I 2 · · · I k of the ideals I 1 , I 2 , . . . , I k is the ideal of R generated by all products of the form x 1 x 2 · · · x k where x i I i for i = 1 , 2 , . . . , k . It follows from the definition of the product of ideals that any element of the product IJ of ideals I and J of a unital commutative ring R can be represented as a sum of the form y 1 z 1 + y 2 z 2 + · · · + y m z m , where y j I and z j J for j = 1 , 2 , . . . , m . Indeed all elements of R representable in this form must belong to the ideal generated by the set { yz : y J and z K } . But the set of elements of R representable as a sum of the above form is itself an ideal of R and is thus the ideal generated by the set of all products yz with y I and z J . 22

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More generally, given ideals I 1 , I 2 , . . . , I k of a unital commutative ring R , the product I 1 I 2 · · · I k of those ideals consists of those elements of R that can be represented as sums of products belonging to the set { x 1 x 2 · · · x k : x i I i for i = 1 , 2 , . . . , k } . Lemma 2.20 A proper ideal P of a unital commutative ring R is prime if and only if, given any ideals I and J of R satisfying IJ P , either I P or J P . Proof Let P be a prime ideal of R . If I 6⊂ P and J 6⊂ P then there exist elements y I and z J such that y 6∈ P and z 6∈ P . Then yz 6∈ P , because the ideal P is prime. But yz IJ . It follows that IJ 6⊂ P . Thus if P is a prime ideal of a unital commutative ring R , and if I and J are ideals of R satisfying IJ P , then either I P or J P . Conversely, suppose that P is a proper ideal of R , and that, for all ideals I and J of R satisfying IJ P , either I P or J P . Let x and y be elements of R satisfying xy P . Then ( x )( y ) = ( xy ), and therefore ( x )( y ) P . It follows that either ( x ) P , in which case x P , or else ( y ) P , in which case y P . This proves that the ideal P is prime. The result follows. 2.6 Unique Factorization Domains The Fundamental Theorem of Arithmetic states that every integer greater than one can be factored uniquely as a product of one or more prime numbers. We now introduce a class of integral domains that possess a unique factor- ization property that generalizes in an appropriate fashion the Fundamental Theorem of Arithmetic.
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