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Proposition B.3.5. [f R is a Noetherian ring and IC R an ideal, then R/I fis Noetherian. Follows forme the canshica( byection preserving lnchusion j deals of R/t} ~— i ideals of R conterning ZL } Fre g(J=S/I AT (shone gi Roe RIL is the Gustent bron) + Theorem B,3.6 (Hilbert basis). [f R is a Noetherian ring, then the poly- Cs) — nomial ring is Noetherian. . ——————— R]Res¥o] —ay Corollary B.3.7. /f R is a Noetherian ring anf(S)is a finitely generated R-algebra, then S is Noetherian. U let Lhe an ideal ta RJ, let Lee fall leading caelfs of elms of Tf = R Claim [js an ileal & R (called the initial icleal of I) I cmtans Oc RK], leading caeft O€R (by ofa) so O€L (cone if G15 =O % supp. not) . Suppor GbE L so J featter, gabrte. in L WLOG suppose ds@. Then f4x%%o is in L aud leading coclt is ath so Lis clesed under + Suppose ACL yo J featte w L Bar any reR_ r{ is in L and leading caeft is TQ so Lis closed under sealer mult by R, A Gee Ris Noetheman by hyp, bs fin gon Cheese generators of L as R-ideal say Gi; Gasiisss a,é€L (neN) ~ then Wee £1, ns ; » choose fi nm LT whose leaching olf is ay «let @& € IN be the depece of fi -let N= mar ( €1,€2,.--n) eN + for each déeW with d Megat polyremrea| -fancton denn oy t Thus, the zeroes of £ are a subset of Aj. at the port P Definition 2.1.4. Given TC R,,, the zero set of T is defined by 7 . the set of commou zeros V9 - [2m {re az: 47) =0 vp eT}. = M45) ne the Pe nT ter When T = {f} for some f € Rn, we will also write] Z(F) in place of Z({f}). pes Proposition 2.2.12. Given subsets T;,T. C Rn , we have: @ NCH AN)24(T). "Zs incluscon -reversing ” w. z(n) = (ze) > (2m) = 2%) teT A FER ‘CBT by agp.) Definition 2.1.4. An algebraic set (of Af) is a subset X of Ay such that there exists TC R,, such that X = Z(T). C Aj Proposition 2.1.6. The collection of algebraic sets (of An) satisfies the conditions to form the closed sets of 0 topoloieel-apage: Specifically: lipology on Aj, (@) Ui 2G) = ZN 2) for any Th...) Tm © Rn- Maat Lepotes the set of products of the form fi-+: fm with fj € 7; for all i. (ii) Nicer AN) = ZUier Ti) for any (possibly infinite) collection of subsets {TC Rn :i€ D} (iii) Z)=0, 2(0) = Ag. "Z turns («rb) union into intersection” (i) Y PEA, PeEUN Zh) if Ajelijm} st. Pe ZT) iff Tjelim} ct, WSET), one hes f(P)=0 fret ETAT, one. hos (fi-+ Sm (P) = filP) = Sn(P) = 0 # Pe 2r,n) arl P@ UM ZT) if Vickie} onehe P ¢2(T) if Victim} Fret suck tet f(P) 40 (kk a Feld ) ih fren TIM ATS such Het (St--Sm)(P) #0 if P¢ 27) (i) Y PEM, Pe Mer ZN) if Viel, omhas Pe ZT) iA Viel, VhET, omlus fi(P)=0 i WSE User Ti, on has f(P)=0 if Pe ZUierTi) (i) Z0)=9, _ we oh A aUb)= I (1) By TH. “plied to GER, , have Z(CuH)) S Ale @ J P=(4,..4) € Ar st. GP)=u(e..a) is #0 in k Then $(P.4) = uP) yt 4-4 clP)y + Col) © Ly] is ¥O in Aly] => hes only fentely many y-wots ink RA k ay cbred) > k whinke , © Tack st. thao wk > (PG) CAL cart m ZE(KY)) co 2) AC 4 Real Suppose FF Is a fuite field FF 2g Thon P(x) = xt-x CKD] is £0 but has every Pell, 6s hot Exercise 2.1.14. X; ~{USRy) < Lhe) (b) Show that Aj is irreducible for all n. #0 ipove. “< bo obser: BP AS ZM)UZ > Th ~~ laden wneers4 ) Show that i Le Ry € Le Ry Sa + ane fy ial, ge OTL telys 4, 4, ten re Kee - v2) +2(Tt) C28) GA, 2! 
