4 MASAYOSHI NAGAT A This theorem ha s a refinement whic h assert s something more o n th e choic e o f th e ele- ments zt\ th e detai l an d th e proo f ar e written i n several textbooks, an d w e omit them . How - ever, we shall state an d prov e severa l corollaries t o thi s normalization theorem . THEOREM 1.11 . Assume that R is an integral domain generated by a finite number of elements over a field k. IfP 0 , P v . . . , Pt are prime ideals such that (i ) P0 = 0 , (ii) Pt is maximal, (iii) P0 C Px C C Pf and (iv) there is no prime ideal Q such that P t _ x C Q C Pt for any i, then t must be the transcendence degree of R over k. PROOF. W e proceed b y inductio n o n t. B y Theorem 1.10 , ther e ar e algebraicall y inde - pendent element s z v . . . , zu ove r k suc h that R i s integral ove r F k[zv . . . , z u ]. I f t = 0, then R i s a field. The n th e integra l dependenc e o f R ove r F implie s that F i s a field , and the n u = 0. Assum e now tha t t 0 . B y the las t statemen t o f Theorem 1.10 , w e may assume tha t z x ^P x . Sinc e R i s integral ove r F, height(Pl n F) = l, 1 an d thi s implies tha t Px n F ztF. The n w e apply ou r inductio n t o R/Px, whic h i s integral ove r FjzxF = k[z2, . . . , zu], an d t o th e prim e ideal s Pi/P1. Thu s we have t - 1 = u - 1 and t = u. Q.E.D. COROLLARY 1.12 . If an integral domain R is finitely generated over a field k, then for any prime ideal P of R, it holds that height P + Krul l di m R/P = Krull di m R = trans, deg k R. COROLLARY 1.13 . If a ring R is finitely generated over a field k and if M is a maxi- mal ideal of R, then R/M is algebraic over k. THEOREM 1.14 . If a ring R is finitely generated over a field k and if I is an ideal of R, then the intersection of all maximal ideals containing I coincides with the radical \/7~ of I (y/l = (the intersection of all prime ideals containing I) = {f^R \f m I for some m}). (Hilbert zero-point theorem) PROOF. I t suffice s t o sho w th e cas e wher e / i s prime. The n considerin g R/I, w e may assume tha t 1=0. Le t / be a nonzero elemen t o f R an d conside r R[l/f] an d it s maximal idealM*. R[l/f] /M f i s algebraic ove r k, an d R/(M' n R) i s algebraic ove r k. Thu s M' n R is a maximal idea l o f R no t containin g /. Q . E. D. Although ther e ar e som e othe r importan t application s o f th e normalization theorem , we omit th e details . This follow s fro m a result whic h assert s tha t i f a n integral domai n R i s integral ove r a normal in - tegral domai n F, then fo r an y ide-a l / o f R, i t hold s tha t heigh t / = heigh t (I n F). Thi s resul t i s a corol- lary t o th e Going-dow n Theorem .
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