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College Algebra Exam Review 301

College Algebra Exam Review 301 - 6.7 NOETHERIAN RINGS 311...

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6.7. NOETHERIAN RINGS 311 Proposition 6.7.2. For a ring R (not necessarily commutative, not neces- sarily with identity), the following are equivalent: (a) Every ideal of R is finitely generated. (b) R satisfies the ascending chain condition for ideals. Proof. Suppose that every ideal of R is finitely generated. Let I 1 I 2 I 3 be an infinite, weakly increasing sequence of ideals of R . Then I D [ n I n is an ideal of R , so there exists a finite set S such that I is the ideal generated by S . Each element of S is contained in some I n ; since the I n are increasing, there exists an N such that S I N . Since I is the smallest ideal containing S , we have I I N [ n I n D I . It follows that I n D I N for all n N . This shows that R satisfies the ACC for ideals. Suppose that R has an ideal I that is not finitely generated. Then for any finite subset S of I , the ideal generated by S is properly contained in I . Hence there exists an element f 2 I n .S/ , and the ideal generated by S is properly contained in that generated by S [ f f g . An inductive
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