By Idun Reiten, Sverre O. Smalø, Øyvind Solberg
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Additional resources for Algebras and Modules Two
Hint: If a = pi' ... p,' :A 0 is an ideal, then choose elements n; E p; gyp? and apply the Chinese remainder theorem for the cosets n; mod p;'". t'.. Exercise 5. The quotient ring o/a of a Dedekind domain by an ideal a 0 is a principal ideal domain. Hint: For a = p" the only proper ideals of o/a are given by p/p" Choose rr E p p"-1/p" p2 and show that p" = orr" + p". Exercise 6. Every ideal of a Dedekind domain can be generated by two elements. C4" Hint: Use exercise 5. ors °'n Exercise 7. In a noetherian ring R in which every prime ideal is maximal, each descending chain of ideals ai 2 a2 ) becomes stationary.
2), we may pick a system a1i ... E OK, ai 0, in such a way that every a E OK with 0 < I NK IQ (a) I < C is associated to one of these numbers. The set N T=SnU X(jai)-1 i=1 then has the required property: since X is bounded, so is X (jai)-1 and therefore also T, and we have S = U Tjs. EEOK Chapter I. Algebraic Integers 42 In fact, if y E S, we find by the above an a E OK, a 0, such that ja E Xy-1, so ja = xy-1 for some x r= X. Since A INKIQ(a)I =IN(xy-1)i =IN(x)I For a prime ideal p of o one always has PO O. In fact, let 7r E P N p2 (p 0), so that fro = pa with p a, hence p+a = o. II. Writing 1=b+s,with bEpand sEa,we find s0pand spcpa =7ro. , s E p, a contradiction. A prime ideal p # 0 of the ring o decomposes in 0 in a unique way into a product of prime ideals, = Tel . . e, . p Instead of pO we will often write simply p. The prime ideals X31 occurring in the decomposition are precisely those prime ideals q3 of 0 which lie over p in the sense that one has the relation p=g3no.
For a prime ideal p of o one always has PO O. In fact, let 7r E P N p2 (p 0), so that fro = pa with p a, hence p+a = o. II. Writing 1=b+s,with bEpand sEa,we find s0pand spcpa =7ro. , s E p, a contradiction. A prime ideal p # 0 of the ring o decomposes in 0 in a unique way into a product of prime ideals, = Tel . . e, . p Instead of pO we will often write simply p. The prime ideals X31 occurring in the decomposition are precisely those prime ideals q3 of 0 which lie over p in the sense that one has the relation p=g3no.