9   Dedekind domains  Definition 9 1   Dedekind domain    A Dedekind domain  is a ring R such that    i  R is a Noetherian integral domain    ii  R is integrally closed in Frac   R      iii  Every non zero prime ideal is maximal   Example  The ring of integers in a number field is a Dedekind domain   Any PID  hence a discrete valuation ring   is a Dedekind domain   Theorem 9 2  A ring R is a discrete valuation ring if and only if R is a Dedekind domain with exactly one non zero prime    Lemma 9 3  Assuming that   R is a Noetherian ring  I   R a non zero ideal  Then  there exists non zero prime ideals p 1       p r such that p 1       p r   I   Proof  Suppose not  Since R is Noetherian  we may choose I maximal with this property  Then I is not prime  so there exists x   y   R   I such that x   y   I   Let I 1     x    I 2   I     y    Then by maximality of I  there exist p 1       p r and q 1       q s  such that p 1   p r   I 1 and q 1   q s   I 2  Then p 1   p r q 1   q s   I 1 I 2   I     Lemma 9 4  Assuming that   R is an integral domain  R is integrally closed in K   Frac   R    0   I   R a finitely generated ideal  x   K  Then  if x I   I  we have x   R   Proof  Let I     c 1       c n    We write      x c i     j   1 n a i j c j     for some a i j   R  Let A be the matrix A     a i j   1   i   j   n and set B   x id n   A   M n   n   K     Then in K n     B   c 1   c n     0       Multiply by adj   B    the adjugate matrix for B   We have      det   B   id n   c 1   c n     0       Hence det   B     0  But det B is a monic polynomial with coefficients in R  Then x is integral over R  hence x   R     Proof of Theorem  9 2         Clear       We need to show R is a PID  The assumption implies that R is a local ring with unique maximal ideal m  Step 1   m is principal   Let 0   x   m  By Lemma 9 3    x     m n for some n   1  Let n  minimal such that   x     m n  then we may choose y   m n   1     x     Set     x y  Then we have y m   m n     x   and hence     1 m   R  If     1 m   m  then     1   R by Lemma 9 4 and y     x    contradiction  Hence     1 m   R  so m     R is principal   Step 2   R is a PID   Let I   R be a non zero ideal  Consider a sequence of fractional ideals I       1 I       2 I     in K  Then since     1   R  we have     k I         k   1   I for all k by Lemma 9 4  Therefore since R is Noetherian  we may choose n maximal such that     n I   R  If     n I   m          then       n   1     R  So we must have     n I   R  and hence I       n       Let R be an integral domain and S   R a multiplicatively closed subset  x   y   S implies x y   S  and also have 1   S   The localisation S   1 R of R with respect to S is the ring      S   1 R     r s   r   R   s   S     Frac   R         If p is a prime ideal in R  we write R   p   for the localisation with respect to S   R   p   Example  p     0    then  R   p     Frac   R     R           p       a b   a         b   p     1    where p is a rational prime   Facts    not proved in this course  but can be found in a typical course   textbook on commutative algebra   R Noetherian implies  S   1 R is Noetherian   Corollary 9 5  Let R be a Dedekind domain and p   R a non zero prime ideal  Then R   p   is a discrete valuation ring    Proof  By properties of localisation   R   p   is a Noetherian integral domain with a unique non zero prime ideal  p R   p     It suffices to show  R   p   is integrally closed in  Frac   R   p       Frac   R    since then  R   p    is a Dedekind domain hence by Theorem 9 2   R   p   is a discrete valuation ring     Let x   Frac   R   be integral over  R   p    Multiplying by denominators of a monic polynomial satisfied by x  we obtain      s x n   a n   1 x n   1       a 0   0       with a i   R  s   S   R   p  Multiply by s n   1  Then x s is integral over R  so x s   R  Hence  x   R   p       Definition 9 6   Valuation on a Dedekind domain    If R  is a Dedekind domain  and p   R a non zero prime ideal  we write v p for the normalised valuation on  Frac   R     Frac   R   p     corresponding to the discrete valuation ring  R   p     Example  R      p     p    then v p is the p adiv valuation    Theorem 9 7  Assuming that   R is a Dedekind domain  I   R a non zero ideal  Then  Ican be written uniquely as aproduct of prime ideals       I   p 1 e 1   p r e r      with p i distinct    Remark   Clear for PIDs  PID implies UFD     Proof  Sketch    We quote the following properties of localisation    i  I   J   I R   p     J R   p   for all prime ideals p    ii  If R a Dedekind domain  p 1   p 2 non zero ideals  then      p 1 R   p 2       p 2 R   p 2   p 1   p 2 R   p 2   p 1   p 2     Let I   R be a non zero ideal  By Lemma 9 3  there are distinct prime ideals p 1       p r such that p 1   1   p r   r   I  where   i   0   Let 0   p be a prime ideal  p     p 1       p r    Then property  ii  gives that   p i R   p     R   p    and hence   I R   p     R   p     Corollary 9 5 gives   I R   p       p i R   p i       i   p i   i R   p i   for some 0     i     i  Thus I   p 1   1   p r   r by property  i    For uniqueness  if I   p 1   1   p r   r   p 1   1   p r   r then   p i   i R   p i     p i   i R   p i   hence a i     i by unique factorisation in discrete valuation rings