Neggers, J.

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STICHTING MATHEMATISCH CENTRUM 2e BOERHAAVESTRAAT 49 AMSTERDAM AFDELING ZUIVERE WISKUNDE 'ZW 1965-003 Quasi-multiplications and inertial automorphisms (I) by J. Neggers Februari 1965 The Mathematical Centre at Amsterdam, founded the 11th of February, 1946, is a non--profit institution aiming at the promotion of pure mathematics and its applications, and is sponsored by the Netherlands Government through the Netherlands Organization for Pure Research (Z. W. O.) and the Central National Council for Applied Scientific Research in the Netherlands (T. N. 0. ) 1, by the Municipality of Amsterdam and by several industries. Quasi-multiplications and inertial automorphisms (I) · J. Neggers The purpose of this note is to introduce a definition of "inertial automorphism" on an arbitrary-commutative ring with unity which reduces to the old definition in the case Risa complete discrete valuation. ring. We generalize the notion of a value non-decreasing mapping o,n a valuation ring to the concept of "quasi-multiplication on a ring". We observe that rings Rare embeddable in the ring of "quasi-multiplications" on R. Using this notion we develop some induced homomorphism theorems (theorems 2 & 3). We define inertial automorphisms as automorphisms which are also quasi-multiplicationso We generalize the notion of "valuation-ring" to M-ring, where Mis a chain of ideals with valuation-like properties. The strong 3rd condition in the definition was needed to give the result of theorem 4. Finally we begin a study of certain classes of subrings defined by the M-structu:r-e and the automorphism structure which have proven important in the case of valuation rings. Rings of quasi-multiplications Suppose Risa commutative ring with identity, then a ring RQ containing Risa ring of quasi-multiplications on R if any ideal of R is an ideal of RQ as well. If R is eL commutative ring with identity and RQ a ring of quasi-multiplications on R9 then y~RQ and xER, implies y(x)c.(x) 9 i.e. 9 * ~· *( ) . yx = ux, uER. If we let f : R -+ R c.e deiinedby f x = u, then it follows that. yx = xf*(x), i.e., we can regard y as a multiplication of x "by a function on R", hence the name quasi-multiplication. Notice that if we define a function f: R-+ R to be a quasi-multi-plication. if there is a function f*: R -+ R such that f(x) = xr"\x), then YE RQ implies "multiplication by y" is a quasi-multiplication. In the situation where Risa valuation-ring with valuation V then a function f: R-+ Risa quasi-multiplication if and only if f is value non-decreasing, i.e., V(f(x)) > V(x) for all x. In this sense we can view quasi-multiplications as natural generalizations of value non-decreasing functions on a valuation ring to arbitrary commutative rings with identity. Lemma 1: :Suppose R is an arbitrary commutative ring with identity, then the collection R [Q] of all quasi-multiplications is a ring under the regular definitions of operator :addition and multiplication. R [QJ has identity I, I(x) = x. Proof: Su:ppose f 9 gE:R[QJ, X€.R 1then* * * * (f + g)(x) = f(x) + g(x) = xf (x) + xg (x) = x(f + g )(x) and f + geR[Q]. * * * Furthermore, (fg){x) = f(g(x)) = g(x)f (g(x)) = xg (x)f (g(x)) and fgE: R [QJ. Lemma 2: If on R[Q} we define (f*g){x) = f(x)g(x), then R[Q] becomes a commutative ring. 2 Proof: * * .. (f * g)(x) = f(x)g(x) = xf (x)g(x) = xf'(x)g (x) and f * g GR [Q]. Since R is commutative it follows that f * g = g * f. We'll denote the ring in lemma 1 by R<Q> and the ring in lemma 2 by R<<Q>>" Lemma 3: If' y eR, let My: R-+ R be defined by My(x) = y(x) • Proof: Then the mapping 4>: R-+ R<Q> defined by 4>(y) = My is an isomorphism.· That 4>(y1 + y2) = 4>(y1) + 4>(y2) is obvious. Next, observe that 4>(y y) = M = M M = 4>(y1)4>(y2). 1 2 Y1Y2 Y1 Y2 Also 4>(y) = 0 implies yx = 0 for all x. Since R has an identity we obtain that ;y1 = y = 0 and 4> is an isomorphism. Theorem 1: Suppose RQ is a ring of quasi-multiplications on R, then RQ can be "embedded" in R<Q>• Proof': Let YE::RQ' then letting f'y: R -+ R be defined by yx we get a mapping 4>: RQ-+ R<Q>" That 4> is a homomorphism is clear. Suppose 4>(y) = 0 9 then yx = 0 for all xe-R. Thus = f (x) y Ker 4> = Annihilator of R in RQ. It is clear that RQ/Ker 4> is _a -ring ·of quasi-multiplications on R (R contains 1, hence ~1,Ker 4> for y # 0!) and on RQ/Ker 4> the mapping constructed above is an isomorphism. From now on we will alwEcys assume that a ring RQ of quasi-multi-plications on R has annihilator (0) so that theorem 1 will hold universally, i.e., any ring RQ of quasi-multiplications will be regarded as a subring of' R<Q> via the natural isomorphism constructed 3 in Theorem 1. Notice that since I= m1 any ring RQ of quasi-multi-plications will also be a ring with identityo Notice that R<Q> .according to the definitions really is,a ring of qu~si-multiplieations •· ' . on Ro Notice further that R<Q> il3 a two-sided R-module, · ioe., · its structure as a left R-module coincides with its structure as a right . R-moduleo This follows from the fact that Risa commutative ring. Thus define (rf){x) = r:f'(x) = f(x)r = (fr)(x). Notice that as a ring operation (fr)(x) = .. _(fmr)(x) = f(rx) ff f(x)r in general! To avoid confusion we shall always use R<Q> as a left R-module. ··.:Theorem 2: If R1, R2 are ·commutative rings with identity and v: R1 ~ R2 is a homomorphism, then v*: R1<Q> + R2<Q> define defined by Proof: * (v (f))(v(y)) = v(f(y)) is a homomorphism into. Ker v is an ideal of R1 thus for any element f ~ R1 <Q> it is true that f(Ker v) C,Ker v, Thus if y E;Ker v, then * * . (v (f))(v(y)) •= (v (f))(O) = v(f(y)) = o. * . Furthermore 9 (v (r1 + r2)){v(y)) = v((f1 + f 2)(y)) = = v(f,(y) + f2(y)) = v(r,(y)) + v(f2(y)) = (v*(f,))(v(y)) + * + (v (f2 ))(v(y)). * Similarly, (v (f1f2))(v(y)) = v((f1f 2)(y)) = v(f,(r2(y))) = * * * = (v (f1))(v(f2(y))) = v (r1)(v (r2)(v(y))) = * * = v (r1)v (r2)(v(y)) • Hence the theorem follows. Theorem .3: If v: R1 + R2 has the property that Ker v c_ x0.r v(x), *. r~ · Proof: then v 1s onto. Indeed, let f: R2 + R2 be a quasi-multiplication. Define f: R1 + R1 as follows. Let f(Ker v) = 0 and if x¢,Ker v, select f(x)'"'-v-\r(v(x))}arbitrarily. We claim that f: R1 + R1 is a quasi-multiplication. 4--. Indeed since f is a quasi-multiplication we have f(v(x)) = v(x) f(v(x)). Thus if' ye,v-1(t(x)), we get V ( y) = V ( X) r ( V ( X) ) = V ( X) V ( Z) = V ( XZ ) o Thus Ye (x) + Ker v = (x} since Ker v c.(x). Hence f'(x) = xr*(x) f'or xf Ker v9 f'(x) = x•O f'or x~Ker v. Thus is f' indeed a quasi-multiplication. By con~truction we get {v*(r))(v(y)}·= v(f'(y)) = f(v(y)), i.e., * - *. v (f') = f' and v is onto. Corollary: If' R1 is a valuation-ring then v* is onto. We are now ready to define the concept of' inertial isomorphism on an arbitrary· commutative ring with identity. Suppose R is such a ring, then an inertial. isomorphism a: R + R is an_isomorphism which is a quasi-multiplication on R. Notice that if' Risa complete valuation ring, then an isomorphism is an inertial isomorphism if' and only if' it is value preserving, i.e., value non-decreasing, i.e., a quasi-multiplication on R. The inertial automorphisms serve as a group of' units in R<Q>' a subgroup of' the group of' units of R<Q>' We shall denote the group of' inertial isomorphisms on R by GI. In the next section we will discuss a type of' ring in which we have the following situation: ( 1) A chain of' ideals 1iii }~ with iii ciii 1 n n=1 i+1 i' (2) 0 iii. = (0) 0 l=W l We'll call this ring an M-ring if' in addition the following condition is satisfied (3} For every X ;. 0 3 an N(x) < co such that ~H~)c (x) I 5 Notice that if R is a valuation ring with value group Z, i.e. 9 A. discrete valu,..t.ion ring and if V(1r) = 1, then letting iii = (1r)n = n = ( 1Tn), M =- {mn} :=1, we get that R is an M-ring. Suppose now that R is an M-ring M = {mn}:;1 , then the M-pseudo-ramification groups Gn are defined as follows: Again notice that then if M = {iii = n G = {oGGI I o(x) - xeiii} • n n if Risa complete discrete valuation ring, (1rn), V('IT) = 1} 9 the M-pseudo-ra.mification groups G are just the ordinary pseudo-ramification group. lrl M=rings and completions Suppose His a commutative ring with identity which is an M-ring with respect to a collection of ideals M = {mn}:=,• Definitio!!.J_: A sequence of functions {fµ}:=, is a null-sequence if given N > 0 .3 µ(N) 3 µ .::_ µ(N) ~: f µ: R -+ ~• Notice that any null-sequence is "eventually" a quasi-multiplication i.e., given x there is a µ such that f (x)e: (x). µ Indeed, s:uppose we take N(x) as in condition ( 3) and pick µ > µ(N(x)), then fµ: R-+ (x) and fµ(x)~(x). ,Next we say: Definition 2: A sequence of functions {fµ}:=, is a limiting sequence if there is a function f such that {fi = fµ - f}:=, is a null-sequence. 6 Proposition 1': If {f' = f - f' }00 and {f" = f - f" }00 are µ µ µ=1 µ µ · µ=1 Proof: · null-sequences 9 then f' = f" o Pickµ!_ µ(N), then fµ - f': R-+ ~ and fJJ - f": R-+ mN (actually µ(N) = max(JJ 1(N), JJ2(N)))o Hence f' - f": R-+ ~• Since this is independent ofµ, we get f' - f": R -+ (\ ii = ( 0) and f' = f". n6.w n Thus limiting sequences have unique limits indicated with lim f. JJ JJ Definition 3: A sequence of functions {f }00_ 1 is Cauchy if given N . JJ JJ-there is a µ(N) such that JJ 19 JJ2 > µ(N) implies f - f R-+ m. .• '1.11 '1.12 J.ll Proposition 2: If a sequence is limiting, then it is Cauchy. Proof: If {fµ}:=l is limiting, suppose limJJfJJ = f and JJ > JJ(N) ~f - f: R -+ ~• Then JJ 1 ,JJ2 > µ(N)~f - f JJ - 00 JJ 1 µ2 = (f f) + (f - f ): R ➔ m.. and {f} _1 is Cauchy. JJ1 µ2 J.ll JJ JJ-The conv,erse is true only under special assumptions on R. Definition 4: A sequence {xJJ}:=l of elements is limiting in case the sequence of functions {f : f (x) = x }00 1 JJ JJ JJ µ= is limiting. Definiticm 5: An M-ring is complete if every Cauchy sequence of constant functions {f : f (x) = x }00 1 is JJ JJ JJ JJ= limiting. Propositj:.£E...l: If Risa complete M-ring 9 then a Cauchy sequence is necessarily limiting. = If R is not complete, then not every Cauchy sequence is limiting. Proof: 7 Suppose Risa complete M-ring, then {fJ.J}:=, a Cauchy-sequence implies {fJ.J(x)}:=, a Cauchy-sequence of elements, hence necessarily limiting. Let f(x) = lim f (x). ® J.J J.J Then {f - f} 1 is a null-sequence and hence lim f = fo J.J J.J= ® J.J J.J If R is not complete, then suppose {x)J.J= 1 is a Cauchy-sequence which is not limitingo Then {f : f (x) = x }® 1 J.J J.J J.J J.J= i1:1 a Cauchy-sequence of functions which is not limiting. We note t,hat if R is an M-ring and RQ is a ring of quasi-multi-plications on R9 then RQ is an M-ring for the same family of ideals M = {m }® of R regarded as ideals of RQo n n=1 Theorem 4: If Risa complete M-ring then R<Q> is also a complete M-ring. Proof: Suppose {fJ.J}:=l is a Cauchy-sequence of quasi-multiplications. Since Risa complete M-ring {fJ.J}:=l is limiting 9 let f = limJ.JfJ.J. Let N(x) be such that ~(x) C (x), then J.J ~ µ(N(x)) ~ ( f - f )( X) G iiiN ( ) C ( X) • Thus f ( X) = f ( X) + XP ( X) = * J.J X * J.J J.J = xf (x) + xp (x) = x(f (x) + P (x)) and f is a quasi-µ J.J J.J J.J multiplication. Thus it follows that R<Q> is a complete M-ring. The Inertial Subring of a Ring Let R be a commutative ring with identity, then let R0 = {x I o(x) = x for all o€GI}. Then we obtain R0 as a subring of R. The inertial subring of R. If mis an ideal of R, then we can construct: 8 v R -+ R/m , let R0 - = R0 /m C R/m. m ;m Proposition 4: If R is an M-ring and R is complete then R0 is complete. Proof: Let {x }°" _1 be a Cauchy-sequence in R0 • Let x = lim x , xG R. JJ JJ- 00 00 JJ JJ Then we have { o( xlJ)} 1,1= 1 = {x) JJ= 1 for all o E GI. Hence x = lim o(x ) • JJ IJ But, o(x - x) = o(x) - o(x )em... if IJ > JJ(N)o IJ IJ N Thus o(x) - x = o(x) - o(x ) + o(x ) - x 4i:..nL. if IJ > JJ(N). IJ JJ IJ .N Since thus o(x) - x E. () iii = (0) we have o(x) = x, Le.• n~w n x ,R0 and R0 is complete. If R is an M-ring say x bR has index of inertia relative to M equal to N if 01(x) - x E: ~ for all o "-Gr but there is a o* such that o(x) - x¢- ~+1• Denote this index by 6M(x). Lemma 4: R0 = {x I 6M(x) = 00 }. Proof: If xeR0 , then o(x) = x ~ mN for all N and 6M(x) = 00 • If 6M(x) = 00 , then o(x) - Xf:i.a mn = (0) and xe:-R0 • Note that the index-of-inertia on R0 is independent of the system M which makes Ran M-ring. Proposition 5: Suppose R is an intergral domain and xis integral over R0 , if oE:GI then o(x)/x is a root of unity in R. If x has degree n over R0 , then o(x) /xis an ,.,!h root f 't · R .  . o uni y in • Proof: * Let K be the qoutient field of Rand K an algebraic closure n n-1 of K. Suppose x + a1 x + • • • + a = O, a ¥ O, a ~ R0 • n n i 9 n We have JI (x-w.) = o, where w.' i = 1 ' 0 0 0 9 n are the l. l. i=1 roots. extend· * * If a€. GI a to a on K 9 then n * JI ( a(x) -a(w.)) = 0 • i=1 1 S:i.nce x = X • XO' 9 XO' a unit in R. * n a (w.) n We get: n JI (x l. ) 0 or JI (x -XO' = i=1 XO' i=1 n n * n Since a 'F o, a = JI w. = JI a (w.) = XO' n n i=1 J i=1 J n JI wj(i) = x/ an• we obtain x/ = 1o i=1 a(x) . th . Thus x = - is an n-- root of unity. O' X n Corollary 1: If x has degree n over R0 , then x ~ R0 • Proof: Suppose x has degree n, then ( a(x))n = 1. X * a (w.) l. ) = XO' * n a (w.) J JI i=1 XO' a(x)n a(xn) n Hence, --- = --~ = 1, i.e., a(x) = n n n x for all a EGI. X X n Thus x e R0 • o. = Corollary _g_: If Risa characteristic O integral domain and D: R ~ R a derivation, then D(R0 ) = 0 =}D(R0 ) = 0 where R0 is the integral closure of R0 in R. Proof: xeR0 implies xneR0 for some n. Thus D(xn) = nxn- 1D(x) = O n-1 ~ ( ) and nx r O implies D x = O. Suppose m cR0 is an ideal• then we define ~ = {xe:R I Xnc:m }. 10 (i) ¼i=\/v (ii) ~ is an ideal in R0 0 . (iii) ' ~= ~ iii•, i,e,, v;; is a radical. ideal VR0Vio m' O o n V, I r--::-' n p - I ,-::::-' x '= _ Vi- m :::!; (x) em ::::!'JX'=Vi_ m R0 0 0 (iv) If mis Suppose Suppose prime, then~ is prime 0 x¢\/; m •• then xne.R0 -==,, xn4;. 0 Ir--=' ( )s - ( )sn xyE. Vi m , then xy E. m ~ xy = 0 t ( snt)( snt) - snt snt If Y E.Ro• then X Y e.m, X 9 Y G.Ro, snt.J_ -X Im. Thus ysnt&. m. and y,:;;. ~ t i.e. 9 V R m' 0 0 is prime. (v) If m is p-primary, then~ is prime. . U iii is P-primary, then ff= P and '1/:W = \j;J is I~ RO 0 But vRVm= V R m •• RO 0 prime. Propositi~: If o c:=.GI' then o/R0 is an inertial automorphism on R0 • Proof: Let xE<R0 , then o(x) = x • x0 with x0 an ~ root of unity. Since 1e=.R0 , we have x0 €,R0 and o/R0 is an inertial auto-morph~sm on R0• 11 Pseudo-inertial subrings of' M-rings Now suppose R is·anM-ring, M = {iiin}:=l" We define rings Noti~e that Rn is indeed a ring. If' x,.r-. Rn' a eGI' then a(x + y) - (x + y) =· (o(x) - x) +-(a(y) - y)G;.m' n o(xy) - xy = a(x)o(y) - xy = a(x)o(y) - a(x)y + a(x)y - xy = = a(x)(a(y) - y) + (o(x) - x)yem • n Lemma 5: R 1iii ; R -:)R +l • n- n n n Proof': o £GI :=,o(mn)c,.mn and X'"-mn_...a(x) - xEmn• Rn ::,Rn+l obviously. Lemma 6: f1 R = R0·• nE:W n Proof': Lemma 12. Lemma 7: If' Risa c9mplete M-ring, then Rn is complete. Proof': Suppose that {xµ}== 1CRn is a Cauchy sequence. Let ati:GI. Thenµ 1 v > µ(N)~µ - x",e~, thus a(x - x ) = a(x ) .:. a(x )E ii._, since OE GI. µ V µ V N Hence {a(x )}~ 1 is Cauchy. We have a(x) = lim a(x ). µ µ= µ µ Select µ0 such that o(x) - o(x )e.iii for allµ>µ • µ n - 0 Then a(x )E.m (x ER !)=]'a(x)cm and xE.R • µ n µ n n n Thus the result follows. We shall call the rings R pseudo-inertial subrings of' R. n n Suppose R is an M-ring, M = {iiin}:=,• Let Gn be the pseudo-inertial groups. 12 Let R[nJ = {x I o(x) = x Voe.on}. Then x,ye:R[nJ, o(x + y) = o(x) + o(y) = x + y, o(xy) = o{x)o(y) = xy. Thus is R[n] a ring. Lemma 8: R[n)CR[n+l]. Proof: Lemma 9: If Risa complete M•ring, then R(n] is complete. Proof.: $Upp6se that {x)·:=1CR[n] is a Cauchy-sequence. •If x = lim x and oeG , then o(x) = lim o(x ) = lim x = x u u n u u u u and x'=R[n]. Thus the lemma follows. Lemma 10: Suppose R is an integral domain and xis integral over R[mJ 9 if o ~G , then o(x) /x is a root of unity in R. Proof: If x ha: degree n over R[m], then o(x) /x is an ~ root of unity in R. The proof is exactly the same as the proof of proposition 5. Corollary .. n 1: If x has degree n over RCmJ, .then x &R[m]. Corollary 2: If Risa characteristic O integral domain and D : R: R a derivation, then D(R[m]) = 0 ::!) : D(R[m]) = O, where R[m] is the integral closure of R[m] in R. Corollary 3: Remark (i) - (vi) of the previous section hold for ideals in R [mJ • Corollary 4: If o &.Gm 9 then o I R [mJ is an inertial automorphism. We shall call the rings R [m] pseudo-inertial subrings of the 2nd kind. 13 References [1] N. Heerema, Derivations on p-adic fields, TAMS 102 ( 1962), PP• 346-351; [2] s. MacLane, Subfields and automo9J:0.sm groups of p-adic fields, Ann. of Math. , 40 ( 1939) , pp. 423-442; [3] J. Neggers, Derivations on p-adic fields (to appear); [4] O.F.G. Schilling, The theory of Valuations, Mathematical Surveys, No. 4, New York, 1950; [5] o. Zariski and P. Samuel, Commutative Algebra, Vols. 1 and 2, D. Van Nostrand Co., Princeton, New Jersey. 

Quasi-multiplications and inertial automorphisms, 1

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