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**Example text**

14, which is also very important in our development of the theory of local analytic geometry. 8, in order to prove that the "normalization of a space" always exists. Before stating the theorem of the finiteness o~ normalization, we recall some facts concerning the discriminant of a polynomial in one variable. 17. Let k be a field (not necessarily algebraically closed), and f = xr + a 1 xr -1 + ... + a r be a polynomial w]th coefficients in k. Consider a field extension k C K in which f splits: Then the discriminant ~ = ~x (I) ~ith respect to the variable x is defined by: A = II(a; - aj).

Hence y. PEl, so that y E I : f2 I : f. It follows that yf E I, and therefore gEl. 11 (Primary Decomposition). Let R be a Noetherian ring and I ~ R be an ideal. Then there exist finitely many primary ideals qI,'" ,qr in R such that I = qi n ... n qr' Proof. Suppose the converse. Then the set d of ideals for which the theorem is not true, that is, d := {I ideal in R : I is not an intersection of finitely many primary ideals}, is nonempty. PI has a maximal element with respect to inclusion. Let I be such a maximal element.

4) This follows from part (2). 7 This means that the leading coefficient of p is 1. 3. The integral closure of R in S is the set of all elements in S which are integral over R. J is reduced, we simply call it the normalization of R. We denote the normalization by R. To use the standard proof from field theory that sum, difference and products of integral elements are integral again we need the assumption that R is Noetherian. 29. 4. Let ReS be rings. Then the integral closure of R in S is a ring.