s13 - John A. Beachy 1 SOLVED PROBLEMS: SECTION 1.3 13. Let...

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Unformatted text preview: John A. Beachy 1 SOLVED PROBLEMS: SECTION 1.3 13. Let P be a prime ideal of the commutative ring R . Prove that if P is a prime ideal of R , then A ∩ B ⊆ P implies A ⊆ P or B ⊆ P , for all ideals A,B of R . Give an example to show that the converse is false. Solution: If P is a prime ideal of R , and A ∩ B ⊆ P , then AB ⊆ P since AB ⊆ A ∩ B , and therefore A ⊆ P or B ⊆ P . In the ring Z 4 , the zero ideal is not a prime ideal. On the other hand, since the ideals of Z 4 form a chain, it is always true that either A ⊆ A ∩ B or B ⊆ A ∩ B . It follows that if A ∩ B = (0), then either A = (0) or B = (0), so the zero ideal of Z 4 provides the desired counterexample. 14. Show that in the polynomial ring Z [ x ], the ideal ( n,x ) generated by n ∈ Z and x is a prime ideal if and only if n is a prime number. Solution: Define φ : Z [ x ] → Z n by φ ( a + a 1 x + ··· + a n x n ) = a , for all polynomials a + a 1 x + ··· + a n x n ∈ Z [ x ]. It follows from Example 1.2.2 that φ is a ring homomor-...
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s13 - John A. Beachy 1 SOLVED PROBLEMS: SECTION 1.3 13. Let...

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