By Nomizu K., Sasaki T.

ISBN-10: 0521441773

ISBN-13: 9780521441773

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**Example text**

Then m = Ax for some x E A. Thus mV = Ax v , v = 0, 1,2, .... If I is a non-zero proper ideal, then, by the Krull intersection theorem, I c m S , I ct. ms+l. Take an element y E 1\ ms+l. , m S C I. So I = mS. Therefore A is a discrete valuation ring. It does not have prime ideals different from o and m, since if m f:. 0, then none of the ideals mk, 2 k;::: 2, is prime. ) We are going to show that A is integrally closed. Now, let an clement u/v of the field of fractions, where u, v E A \ 0, be integral over A.

If G E A[X] and G 2 is a divisor of P, then G is a common divisor of P and pI; hence (1) ==> (2). Conversely, it follows from the condition (2) that P and pI have a common irreducible divisor, say G. Thus P = GH and pi = GF for some F, HE A[X]. Hence G' H = G(F - H'). Therefore G divides Hand G 2 divides P. Let K be the field offractions for the ring A, and let L be an extension of K such that P can be factorized in L[X] into linear factors: P = (X - (1) ... (X - (n). Then condition (3) means that PI((r) = 0 for some r.

Its degree (which e l n ) Recall that if el, ... 16), then we have a natural bijection

### Affine differential geometry. Geometry of affine immersions by Nomizu K., Sasaki T.

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