By F. M. Hall

ISBN-10: 0521070554

ISBN-13: 9780521070553

ISBN-10: 0521298628

ISBN-13: 9780521298629

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**Example text**

Robson, Hereditary Noetherian prime rings. J. Algebra 16, 86–104 (1970) 33. H. Fitting, Über den Zusammenhang zwischen dem Begriff der Gleichartigkeit zweier Ideale und dem Äquivalenzbegriff der Elementarteilertheorie. Math. Ann. 112, 572–582 (1936) 34. M. A. J. Papick, Prüfer Domains, Monographs and Textbooks in Pure and Applied Mathematics, vol. 203 (Marcel Dekker, New York, 1997) 35. M. Fossum, Maximal orders over Krull domains. J. Algebra 10, 321–332 (1968) 36. M. Fossum, The Divisor Class Group of a Krull Domain (Springer, Berlin, 1973) 37.

The following examples show that we cannot admit decomposed primes in Theorem 2, neither when |Po(K)| = 1, nor when |Po(K)| ≤ 2. √ Example 1 Let K = Q( −31). 5]). On the other hand, 5OK = pq where p and q are not principal (there are no integers of OK with norm 5). Consequently, 5 is irreducible in OK and the order of the classes of p and q is 3 (the class number of K is 3). In other words, p3 = π OK and q3 = π OK where π and π are irreducible. Finally, we have 53 OK = π π OK with 3 irreducible elements on the left side and 2 on the right side.

I. J. Algebra 44, 319–338 (1977) 31. M. Deuring, Algebren (Springer, Berlin, 1935) (Revised, 1968) 32. D. C. Robson, Hereditary Noetherian prime rings. J. Algebra 16, 86–104 (1970) 33. H. Fitting, Über den Zusammenhang zwischen dem Begriff der Gleichartigkeit zweier Ideale und dem Äquivalenzbegriff der Elementarteilertheorie. Math. Ann. 112, 572–582 (1936) 34. M. A. J. Papick, Prüfer Domains, Monographs and Textbooks in Pure and Applied Mathematics, vol. 203 (Marcel Dekker, New York, 1997) 35. M.

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