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Some Types of Simple Ring Extensions HOUSTON JOURNAL OF MATHEMATICS, Volume 1, No. 1, 1975. SOMETYPES OF SIMPLERING EXTENSIONS1 MasayoshiNagata Throughout the present article, we understand by a ring a commutative ring with identity, and by an over-ring of a ring R a ring containing R and having the same identity with R. Let T be an over-ringof a ring R. If T is generatedby a singleelement over R and if •v is a homomorphismof T into a ring T', then •vT is generatedby a singleelement over •vR. In õ 1, we discusssome results related to the converse of this fact, and we mainly concern with the case where T is integral over R. In õ2, we deal with a special type of ring extensions which has some algebro-geometricmeaning in connection with birational correspondences,and our resultsinclude a criterion for such an extension to be generatedby a singleelement. The writer wants to express his thanks to ProfessorH. Hijikata and ProfessorW. Heinzer for their discussionswith the writer on the topics in the present article. He is also grateful to the referee for the critical reading of the manuscript. õ 1. Integrally dependent case. Obviously, the following holds: LEMMA 1.1 Let A be an ideal of an over-ring T of a ring R. If T is generated by a singleelement over R, then so is T/A over R/(A ChR). The converse of this is obviously false and we want to discusssome caseswhere the converseis true. We shall begin with the case where R is a field and then we shall discussits generalizationto the casewhere R is a quasi-semi-localring. THEOREM 1.2. Let R be a fieM and assume that its over-ring T is a noetherian local ring with maximal ideal M and that T is integral over R. Then: (1) Assumethat TIM is a separablealgebraic extension of a finite degreeover R, then T is generated by a single element over R if and only if M is principal. 1 Theresults in this article were published already inthe proceedings ofthe 1974 symposium in algebra(ring theory and numbertheory) at Matsuyama.But the proceedingswere written in Japaneseand copiesof the proceedingswere distributed only in Japan,and the writer is willingto publishan Englishversion of the article. 131 132 MASAYOSHI NAGATA (2) Assumethat TIM is an algebraicextension of R which is not separable.If T is generated by a single element over R, then TIM is obviously generated by a single element, say a*, over R and for any representativea (C T) of a*, T is generatedby a over R. PROOF. (1)' The only if part is obvious becauseR[X] is a principal ideal ring. Assumethat M = sT and the T/M = R(c*) with a separableelement c*. Let fiX) = X n + aI Xn-1 + ...+ a n be the minimal polynomial forc* over R and let c be a representative of c*. Then f(c) CM. Let f'(X) be the derivativeof fiX). Sincec* is separable,f'(c) is not in M. (i) Assume first that f(c) generates M. Then we see that R[c] is a dense subset of T, [1]. Since natural topology of T is discrete, we see that R[c]= T. (ii) Assumenow that f(c) doesnot generateM. Thenf(c)C M 2. f(c+s)=f(c) + sf'(c) moduloM 2 andwe see that f(c+s) generates M. Thus, by (i) above,we see that T is generatedby c+s. (2)' Assume that T=R[b] and let b* be (b modulo M). Obviously T/M= R[b*]. Since T/M= R[a*], there is a polynomial g(X)•R[X] such that g(a*) = b*. If it holds that Rig(a)] = T, then we surely have R[a] = T. Thus we may assumethat a* = b*. Let f(X) be the minimal polynomial for b* over R. Since b* is not separable,we havef(a)= fib) moduloM 2. SinceT = Rib], wehave M = f(b)T. Thereforef(a) - fib)C M 2 impliesthat f(a)T = M, andtherefore we see that R[a] = T. q.e.d. THEOREM 1.3. Assumethat a noetheriansemi-local ring (T, M1 ..... M r) is integralover its subfield R.Set T i = TMi(i = ] .....r). Then.' (1) If T isgenerated by a singleelement over R, theneach T i isalso generated by a singleelement over R andT is thedirect sum of T1 ..... T r (2) Assumeconversely that eachT i is generatedby a singleelement b i overR. ThenT is the directsum of T] .....T r (3) Underthe assumptionin (2) above,set b = b] q-...q-b r, b*i =(bi modulo MiTi).Let f•(X) bethe minimalpolynomial for b* i overR. ThenT =Rib] if aridonly if f l(X) .....fr{X) aremutually different from each other. PROOF. (1) and (2): Under the assumption, T is an Artin ring and the assertionsfollow easily. (3): Let ei be theidentity of Ti. Then1 = e1 + ... + er Sincethe maximalideal SIMPLE RING EXTENSIONS 133 ofT i isnilpotent, there is a naturalnumber n such that eifi(bi )n,= 0 forall i = 1,...,r. Then,setting g(X) = (IIifi(x))n,we have g(b)= 0, whichshows that (the number of the maximal ideals of R[b] ) • (the number of mutually distinct polynomials among fl (X),...,fr(X)).Thus the only if partis provectConversely, assume that fl (X),.. ,fr(X) aremutually different from each other Set gi(X! = (IIj_•ifj(X)) n. Thenwe have ejgi(bj!= 0 forj -• i, andeigi(b i) • Miti Thusgi(b)= eigi(bi) which isin W i outside of MiTi Thenthere is an hi(X) C R[X] suchthat eihi(bi) is theinverse of eigi(bi)= gi(b) in Ti Then,with ki(X?= gi(X)hi(X), we see that ki(b,)= ei. ThusRib] containsallof ei andwe seethat R[b] = T. q e d COROLLARY 1.4 Assume that R is a field containing infinitely many elements andthat an over-ringT is thedirect sum of localrings T! .....T r andis integral over R. ThenT is generatedby a singleelement over R if andonly if eachT i (i = I .....r) is generated by a single element over R. Note that these results 1,2 '•, L 4 can be generalized easily to the casewhere R is the direct sum of a finite number of fields in view of the following easy fact' LEMMA 1.5 Assumethat a ringR is the directsum of ringsR I .....R m with identitieseI .....e m respectively.Let T be an over-ringof R. ThenT isgenerated by a singleelement over R if and only if everyeiT (i=!..... tn) is generatedby a single elementover R i. In view of the lemma of Krull-Azumaya (which may be called the lemma of Nakayama), we see immediately the following LEMMA 1.6. Assume that R is a ring with Jacobson radical d and that T is an over-ring which is finitely generated as an R-module. Then T is generated by a single element over R if and only if T/dT is generatedby a singleelement over RId. Note here that our assumptionthat T is integral over R and is a finitely generated module over R is important in view of the fact that there is a discretevaluation ring V of a field such that(i) V is not complete and (ii) the completion of V is integral over V (cf. [ 1 ], pp. 205-207). Anyway, this Lemma 1.6 enables us to generalize our results 1.2 •, 1.4 to the casewhere R is a quasi-semi-localring. õ2. Some results on UFD. By a prime element of a ring, we understandan element which generates a non-zero prime ideal. The term UFD stands for unique 134 MASAYOSHI NAGATA factorization domain. The following result is well known and sometimes called a theorem of Nagata: LEMMA 2.1. Let S be a multiplicativelyclosed subset generated by someprime elements in an integral domain R, satisfying the ascending chain condition for principalideals. If thering R S isa UFD, thenR isalso a UFD. We review here a well known fact on monoidal dilations as follows: LEMMA 2.2. Assumethat a prime ideal P of a noetherianintegral domain R is generatedby s elementsa = a1 ..... a s withs = heightP. SetT = R[a2/a..... as/a]. Then (i) thereis one and only one prime idealP' of heightone in Tso that P' (hR = P, (ii) V= TtY is a discretevaluation ring and P'V = aV, (iii) V is uniquelydetermined, independentlyof the choice of thea i and(iv) if Wis a valuationring of thefield K of fractions of R such that the maximal ideal M of W lies over P (R C_W) and such that a2/a..... as/a modulo M arealgebraically independent over R/(M ClR), thenW coincides with V. The valuation ring V given above is called the divisorof K obtained by the monoidal dilatation with center P. PROOF.We mayemploy Rp insteadof R hencewe mayassume that R is a regularlocal ring. Then the proof is easy and we omit the detail (cf. [ 1], õ38). q.e.d. From now on, we assume HYPOTHESIS2.3. R is a UFD,K isits field of fractionsand Pl,'",Pn are prime elementsof R whichgenerate mutually different ideals. Set f = p1,"',Pn' V1,"',Vn are valuation rings of K satisfying the following four conditions. (i) R iscontained inevery Vi, (ii) PiVi is the maximal ideal of Vi, (iii) pj½ PiVi if i --7:j and (iv) PiVi f3R --7:pi R. Foreach t = 1,...,n,we set R t =R[f '1 ] ClV 1 Cl ... •3 Vt. THEOREM2.4. EachR t is a UFD and the set Ut of unitsin Rt is theset of z t-t-1 z n elementsof the form uPt_t_1 "'Pn with a unit u of R and rationalintegers zt_t_1....
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