Nice derivations over principal ideal domains Nikhilesh Dasgupta and Neena Gupta Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India e-mail: [email protected] 7 1 e-mail : [email protected], [email protected] 0 2 n Abstract a J In this paper we investigate to what extent the results of Z. Wang and D. Daigle 3 on “nice derivations” of the polynomial ring k[X,Y,Z] over a field k of characteristic 1 zero extend to the polynomial ring R[X,Y,Z] over a PID R, containing the field of ] rational numbers. One of our results shows that the kernel of a nice derivation on C k[X ,X ,X ,X ] of rank at most three is a polynomial ring over k. A 1 2 3 4 Keywords. Polynomial Rings, Locally Nilpotent Derivation, Nice Derivation. . h 2010 MSC. Primary: 13N15; Secondary: 14R20, 13A50. t a m [ 1 Introduction 1 v By a ring, we will mean a commutative ring with unity. Let R be a ring and n(> 1) 5 be an integer. For an R-algebra A, we use the notation A = R[n] to denote that A is 3 6 isomorphic to a polynomial ring in n variables over R. We denote the group of units 3 of R by R∗. 0 Let k be a field of characteristic zero, R a k-domain, B := R[n] and m is a positive . 1 integer ≤ n. In this paper, we consider locally nilpotent derivations D on B, which 0 7 satisfy D2(Ti) = 0 for all i ∈ {1,...,m} ⊆ {1,...,n} for some coordinate system 1 (T ,T ,...,T ) of B. For convenience, we shall call such a derivation D as a quasi- 1 2 n : v nice derivation. In the case m = n, such a D is called a nice derivation (Thus a nice i derivation is also a quasi-nice derivation). We investigate the rank of D when n = 3 X and R is a PID (see Section 2 for the definition of rank of D). r a The case when B = k[3] was investigated by Z. Wang in [14]. He showed that rank D is less than 3 for the cases (m,n) = (2,3),(3,3) and that rank D = 1 when D is a nice derivation (i.e., for (m,n) = (3,3)). In [6], Daigle proved that the rank of D is less than 3 even in the case (m,n) = (1,3) ([6, Theorems 5.1 and 5.2]). Now let R be a Noetherian domain containing Q, say R is regular. It is natural to ask how far we can extend the above results to R[X,Y,Z](= R[3]). In particular, we consider the following question for nice derivations. Question 1. If D is a nice derivation of R[X,Y,Z], then is rank D = 1, or, at least, is rank D < 3? 1 In Section 3, we show that when R is a PID, the rank of D is indeed less than 3 (Theorem 3.6) and construct a nice derivation D over k[1] with rank D = 2 (Example 3.5). Moreover, we construct a nice derivation D over k[2] with rank D = 3 (Example 3.10)showingthatTheorem3.6doesnotextendtotwo-dimensional regularorfactorial domains. An important open problem in Affine Algebraic Geometry asks whether the kernel of any D ∈ LND (k[4]) is necessarily finitely generated. In the case when rank D ≤ 3, k Bhatwadekar and Daigle had shown that the kernel is indeed finitely generated [3, Theorem 1]. However Daigle and Freudenburg had constructed an example to show that the kernel need not be k[3] [8]. Under the additional hypothesis that the kernel is regular, Bhatwadekar, Gupta and Lokhande showed that the kernel is indeed k[3] [5, Theorem 3.5]. A consequence of Theorem 3.6 of our paper is that in the case rank D ≤ 3, the kernel of any nice derivation D is necessarily k[3] (Corollary 3.8). Thefollowing questiononquasi-nicederivations arises inviewofWang’s resultthat rank D is less than 3 for (m,n) = (2,3). Question 2. If D is a locally nilpotent derivation of R[X,Y,Z], such that D is irre- ducible and D2X = D2Y = 0, is then rank D < 3? In Section 4, we investigate this question and obtain some partial results when R is a PID (Proposition 4.4) and a Dedekind domain (Proposition 4.6). Example 4.5 shows that Question 2 has a negative answer in general, even when R is a PID. We shall also construct a strictly 1-quasi derivation (defined in Section 4) on R[3] over a PID R (Example 4.9). By a result of Daigle (quoted in Section 4 as Theorem 4.8), there does not exist such a derivation on k[3], where k is a field of characteristic zero. 2 Preliminaries For a ring A and a nonzerodivisor f ∈ A, we use the notation A to denote the f localisation of A with respect to the multiplicatively closed set {1,f,f2,...}. Let A ⊆ B be integral domains. Then the transcendence degree of the field of fractions of B over that of A is denoted by tr.deg B. A A subring A⊆ B is defined to be factorially closed in B if, given nonzero f,g ∈ B, theconditionfg ∈ Aimpliesf ∈ Aandg ∈ A. WhentheambientringB isunderstood, we will simply say that A is factorially closed. A routine verification shows that a factorially closed subring of a UFD is a UFD. If A is a factorially closed subring of B, then A is algebraically closed in B; further if S is a multiplicatively closed set in A then S−1A is a factorially closed subring of S−1B. Let k be a field of characteristic zero, R a k-domain, and B an R-domain. The set oflocallynilpotentR-derivationsofB isdenotedbyLND (B). WhenRisunderstood R from the context (e.g. when R = k), we simply denote it by LND(B). We denote the kernel of a locally nilpotent derivation D by Ker D. Let D ∈ LND (B) and A := Ker D. It is well-known that A is a factorially R closed subring of B [7, 1.1(1)]. For any multiplicatively closed subset S of A\{0}, D extends to a locally nilpotent derivation of S−1B with kernel S−1A and B∩S−1A= A [7, 1.1(2)]. Moreover if D is non-zero, then tr.deg B = 1 [7, 1.1(4)]. A locally A nilpotent derivation D is said to be reducible if there exists a non-unit b ∈ B such that 2 DB ⊆ (b)B; otherwise D is said to be irreducible. An element s ∈ B is called a slice if Ds = 1, and a local slice if Ds ∈ A and Ds 6= 0. Moreover D is said to be fixed-point free if the B ideal (DB)= B. When B := R[n] and D ∈ LND (B), the rank of D, denoted by rank D, is defined R to be the least integer i for which there exists a coordinate system (X ,X ,...,X ) of 1 2 n B satisfying R[X ,...,X ] ⊆ A. i+1 n NowletB beak-domainandDanelementofLND(B)withalocalslicer ∈ B. The Dixmiermapinducedbyrisdefinedtobethek-algebrahomomorphismπ : B → B , r Dr given by (−1)i ri π (f)= Dif . r X i! (Dr)i i>0 The following important result is known as the Slice Theorem [10, Corollary 1.22]. Theorem 2.1. Let k be a field of characteristic zero and B a k-domain. Suppose D ∈ LND(B) admits a slice s ∈ B, and let A = Ker D. Then (a) B = A[s] and D = ∂ . ∂s (b) A= π (B) and Ker π = sB. s s (c) If B is affine, then A is affine. The following theorem of Daigle and Freudenburg characterizes locally nilpotent derivations of R[2], where R is a UFD containing Q [7, Theorem 2.4]. Theorem 2.2. Let R be a UFD containing Q with field of fractions K and let B = R[X,Y] =R[2]. For an R-derivation D 6= 0 of B, the following are equivalent: (i) D is locally nilpotent. (ii) D = α(∂F ∂ − ∂F ∂ ), for some F ∈ B which is a variable of K[X,Y] satisfying ∂Y ∂X ∂X ∂Y gcd (∂F, ∂F) = 1, and for some α ∈ R[F]\{0}. B ∂X ∂Y Moreover, if the above conditions are satisfied, then Ker D = R[F]= R[1]. With the same notation as above, the following lemma gives interesting results when D satisfies some additional hypothesis [14, Lemma 4.2]. Lemma 2.3. Let R be a UFD containing Q, B = R[X,Y](= R[2]) and D ∈LND (B) R such that D is irreducible. Then the following hold: (i) If D2X = 0, then Ker D = R[bY + f(X)], where b ∈ R and f(X) ∈ R[X]. Moreover, DX ∈ R and DY ∈ R[X]. (ii) If D2X = D2Y = 0, then D = b ∂ −a ∂ for some a,b ∈ R. Moreover, ∂X ∂Y Ker D = R[aX +bY]. (iii) If R is a PID and D2X = D2Y = 0, then D has a slice. Over a Noetherian domain containing Q, a necessary and sufficient condition for the kernel of a nonzero irreducible D ∈ LND (R[X,Y]) to be a polynomial ring is R given by [4, Theorem 4.7]. 3 Theorem 2.4. Let R be a Noetherian domain containing Q and let D be a non-zero irreducible locally nilpotent derivation of the polynomial ring R[X,Y]. Then the kernel A of D is a polynomial ring in one variable over R if and only if DX and DY either form a regular R[X,Y]-sequence or are comaximal in R[X,Y]. Moreover if DX and DY are comaximal in R[X,Y], then R[X,Y] is a polynomial ring in one variable over A. An importantresulton fixed-pointfreelocally nilpotent derivations is the following [10, Theorem 4.16]. Theorem 2.5. Let R be any Q-algebra, and let B = R[X,Y] = R[2]. Given D ∈ LND (R[X,Y]), the following conditions are equivalent: R (1) D is fixed-point free, i.e., (DB) = B, where (DB) is the B-ideal generated by DB. (2) There exists s ∈ B with Ds = 1. In addition, when these conditions hold, Ker D = R[1]. For a ring containing Q, the following cancellation theorem was proved by Hamann [11, Theorem 2.8]. Theorem 2.6. Let R be a ring containing Q and A be an R-algebra such that A[1] = R[2]. Then A = R[1]. The following is a well-known result of Abhyankar, Eakin and Heinzer [1, Proposi- tion 4.8]. Theorem 2.7. Let C be a UFD and let X ,...,X be indeterminates over C. Suppose 1 n that A is an integral domain of transcendence degree one over C and that C ⊆ A ⊆ C[X ,...,X ]. If A is a factorially closed subring of C[X ,...,X ], then A= C[1]. 1 n 1 n Thefollowing local-global theorem was proved by Bass, Connell and Wright [2] and independently by Suslin [13]. Theorem 2.8. Let R be a ring and A a finitely presented R-algebra. Suppose that for all maximal ideals m of R, the R -algebra A is isomorphic to the symmetric algebra m m of some R -module. Then A∼= Sym (L) for some finitely presented R-module L. m R The following result is known as Serre’s Splitting Theorem [12, Theorem 7.1.8]. Theorem 2.9. Let A be a Noetherian ring of finite Krull dimension. Let P be a finitely generated projective A-module of rank greater than dimension of A. Then P has a unimodular element. Following isthefamousCancellation TheoremofHymanBass[12,Theorem7.1.11]. Theorem 2.10. Let R be a Noetherian ring of dimension d and P a finitely generated projective R-module of rank > d. Then P is “cancellative”, i.e., P ⊕Q ∼= P′ ⊕Q for some finitely generated projective R-module Q implies that P ∼= P′. We now state a local-global result for a graded ring [12, Theorem 4.3.11]. 4 Theorem 2.11. Let S = S ⊕ S ⊕ S ... be a graded ring and let M be a finitely 0 1 2 presented S-module. Assume that for every maximal ideal m of S , M is extended 0 m from (S ) . Then M is extended from S . 0 m 0 For convenience, we state below an elementary result. Lemma 2.12. Let A and B be integral domains with A ⊆ B. If there exists f in A, such that A = B and fB∩A= fA, then A = B. f f Proof. Let b ∈ B. Suppose, if possible b ∈/ A. Now since B = A , we have b ∈ A . f f f Hence there exist a ∈ A and an integer n > 0 such that b = a/fn. We may assume that n is the least possible. But then a ∈ fB∩A = fA. Let a = fa for some a ∈ A. 1 1 Then b = a /fn−1, contradicting the minimality of n. 1 3 Nice Derivations In this section, we shall explore generalisations of the following theorem of Z. Wang [14, Proposition 4.6]. Theorem 3.1. Let K be a field of characteristic zero and K[X,Y,Z] = K[3]. Suppose that D(=6 0) ∈ LND (K[X,Y,Z]) satisfies D2X = D2Y = D2Z = 0. Then the K following hold: (i) Ker D contains a nonzero linear form of {X,Y,Z}. (ii) rank D = 1. ′ ′ ′ (iii) If D is irreducible, then for some coordinate system (X ,Y ,Z ) of K[X,Y,Z] related to (X,Y,Z) by a linear change, ′ ∂ ′ ∂ D = f(X ) +g(X ) ′ ′ ∂Y ∂Z ′ where f, g ∈ K[X ] and gcd ′ (f,g) = 1. K[X ] We first observe the following result. Lemma 3.2. Let R be a UFD containing Q and D(6= 0) ∈LND (R[X,Y,Z]), where R R[X,Y,Z] = R[3] and rank D < 3. Then Ker D = R[2]. ′ Proof. Let A := Ker D. Since rank D < 3, there exists X ∈ R[X,Y,Z] such that R[X,Y,Z] = R[X′][2] and X′ ∈ A. Then taking C = R[X′], it follows from Theorem 2.7 that A(= Ker D)= R[X′][1] =R[2]. The following example shows that Lemma 3.2 does not extend to a Noetherian normal domain R which is not a UFD. Example 3.3. Let R[a,b] = R[2] and R := R[a,b] . Let B := R[X,Y,Z] = R[3] and (a2+b2−1) D be an R-linear LND of B, such that DX = a, DY = b−1 and DZ = aY +(1−b)X. 5 Setting u = aY +(1−b)X, v = (1+b)Y +aX and w = 2Z +uY −vX, we see that Du= Dv = Dw = 0 and D2X = D2Y = D2Z = 0. Let A := Ker D. Now B(1+b) = R(1+b)[v,w,X] and B(1−b) = R(1−b)[u,w,Y]. Thus it follows that A(1+b) = R(1+b)[v,w] = R(1+b)[2] and A(1−b) = R(1−b)[u,w] = R(1−b)[2]. Since (1+b) and (1−b) are comaximal elements of R, A =R[u,v,w] and A = R [2] m m for every maximal ideal m of R. Now B = R[X,Y,Z] = R[X,Y,w] and w ∈ A; so rank D < 3. Setting T = u, we a see that A= R[aT,(1+b)T,w]. By Theorems 2.8 and 2.9, A= Sym (F ⊕P), where R F is a free R-module of rank 1 and P is a rank 1 projective R-module given by the ideal (a,1+b)R, which is not principal. Hence P is not stably free and so A 6=R[2] [9, Lemma 1.3]. Remark 3.4. In Proposition 3.9, we will see that over any Dedekind domain R, the kernel of a nice derivation of R[3] is generated by (at most) three elements. The following example shows that Part (ii) of Theorem 3.1 does not hold when K is replaced by a PID R. Example 3.5. Let k be a field of characteristic zero, R = k[t] = k[1] and B := R[X,Y,Z](= R[3]). Let D ∈ LND (B) be such that R DX = 0, DY = X −t and DZ = X +t. Let A = Ker D and G := (X −t)Z −(X +t)Y. We will show that (i) A= R[X,G]. (ii) B 6= A[1]; in fact, B is not A-flat. (iii) rank D = 2. Proof. (i) Let C := R[X,G]. We show that C = A. Clearly C ⊆ A. Set f := X −t. Then B = R[X,G,Y] = C [1]. Hence, as both C (⊆ A ) and A are factorially f f f f f f closed subrings of B and as tr.deg B = 1= tr.deg B, we have C = A . f C f A f f f Now B/fB may be identified with R[Y,Z](= R[2]). Clearly C/fC = R[1] and the image of C/fC in B/fB is R[tY](= R[1]). Thus the natural map C/fC → B/fB is injective, i.e,fB∩C = fC. SinceAisfactoriallyclosedinB,wealsohavefB∩A= fA and hence fA∩C = fB ∩A∩C = fB ∩C = fC. Therefore as C = A , we have f f C = A by Lemma 2.12. (ii) (X −t,X +t)B is a prime ideal of height 2 in B and (X −t,X +t)B ∩A = (X,t,G)A is a prime ideal of height 3 in A, violating the going-down principle. Hence B is not A-flat and therefore B 6= A[1]. (iii)SinceDX = 0,rank D < 3. Ifrank D = 1,thenclearlyB = A[1] contradicting (ii). Hence rank D = 2. We now prove an extension of Theorem 3.1 over a PID. Theorem 3.6. Let R be a PID containing Q with field of fractions L and B := R[X,Y,Z] = R[3]. Let D(6= 0) ∈ LND (B), and A := Ker D. Suppose that R D is irreducible and D2X = D2Y = D2Z = 0. Then there exists a coordinate system (U,V,W) of B related to (X,Y,Z) by a linear change such that the following hold: 6 (i) A contains a nonzero linear form of {X,Y,Z}. (ii) rank D ≤ 2. In particular, A = R[2]. (iii) A = R[U,gV − fW], where DV = f, DW = g, and f,g ∈ R[U] such that gcd (f,g) = 1. R[U] (iv) Either f and g are comaximal in B or they form a regular sequence in B. More- over if they are comaximal, (i.e., D is fixed-point free) then B = A[1] and rank D = 1; and if they form a regular sequence, then B is not A-flat and rank D = 2. Proof. (i) D extends to an LND of L[X,Y,Z] which we denote by D. By Theorem ′ ′ 3.1 there exists a coordinate system (U,V ,W ) of L[X,Y,Z] related to (X,Y,Z) by a linear change and mutually coprime polynomials p(U), q(U) in L[U] for which ∂ ∂ D = p(U) +q(U) . ′ ′ ∂V ∂W MultiplyingbyasuitablenonzeroelementofR,wecanassumeU ∈ R[X,Y,Z]. Clearly A = KerD∩R[X,Y,Z]andU ∈ A. Moreover withoutlossofgenerality wecanassume that there exist l,m,n ∈ R with gcd (l,m,n) = 1 such that U = lX +mY +nZ. As R R is a PID, (l,m,n) is a unimodular row of R3 and hence can be completed to an U X     invertible matrix M ∈ GL (R). Let V =M Y . 3 W Z Then R[U,V,W] = R[X,Y,Z] and as U ∈ A, A contains a nonzero linear form in X,Y,Z. (ii) Follows from (i) and Lemma 3.2. (iii) R[U] is a UFD and B = R[U,V,W] = R[U][2]. So D is a locally nilpotent R[U]-derivation of B. Now the proof follows from Part (ii) of Lemma 2.3. (iv) Since B = R[U,V,W] = R[U][2], the first part follows from Theorem 2.4. Moreover when f and g are comaximal in B, it also follows from Theorem 2.4 that B = A[1]. Hence in this case rank D = 1. If f and g form a regular sequence in B (and hencein Asince Ais factorially closed in B), (f,g)B ∩A = (f,g,gV −fW)A. But (f,g,gV −fW)A is an ideal of height 3, while (f,g)B is an ideal of height 2, violating the going-down principle. It follows that in this case B is not A-flat. In this case indeed rank D = 2, or else if rank D = 1, we would have B = A[1]. The proof of Theorem 3.6 shows the following: Corollary 3.7. With the notation as above, the following are equivalent: (i) B = A[1]. (ii) rank D = 1. (iii) B is A-flat. Proof. (i)⇔(ii) and (ii)⇒(iii) are trivial. (iii)⇒(i) follows from Theorem 3.6(iv). As mentioned in the Introduction, Theorem 3.6 shows that the kernel of an irre- ducible nice derivation of k[4] of rank ≤ 3 is k[3]. More precisely, we have: 7 Corollary3.8. LetK beafieldofcharacteristic zeroandletK[X ,X ,X ,X ]= K[4]. 1 2 3 4 Let D ∈ LND (K[X ,X ,X ,X ]), be such that D is irreducible and DX = 0 and K 1 2 3 4 1 D2X = 0 for i = 2,3,4. Then Ker D = K[3]. i By a result of Bhatwadekar and Daigle [3, Proposition 4.13], we know that over a Dedekind domain R containing Q, the kernel of any locally nilpotent R-derivation of R[3] is necessarily finitely generated. We now show that if D is a nice derivation, then the kernel is generated by at most three elements. Proposition 3.9. Let R be a Dedekind domain containing Q and B := R[X,Y,Z] = R[3]. Let D ∈ LND (B) such that D is irreducible and D2X = D2Y = D2Z = 0. Let R A := Ker D. Then the following hold: (i) A is generated by at most 3 elements. (ii) Moreover, if D is fixed-point free, then rank D < 3 and D has a slice. In particular, rank D = 1. Proof. (i) By Theorem 3.6, A = R [2] for all p ∈ Spec(R). Hence by Theorem 2.8, p p A ∼= Sym (Q) for some rank 2 projective R-module Q. Since R is a Dedekind domain R by Theorem 2.9, Q ∼= Q ⊕M where Q is a rank 1 projective R-module and M is a 1 1 free R-module of rank 1. Again since R is a Dedekind domain Q is generated by at 1 most 2 elements. Hence A is generated by at most 3 elements. (ii) Now assume D is fixed-point free. Let DX = f , DY = f and DZ = f . 1 2 3 Then, by Theorem 2.1, B = A [1] for each i ∈ {1,2,3}. Since (f ,f ,f )B = B fi fi 1 2 3 we have B = A [1], for each p˜ ∈ Spec(A). Hence, by Theorem 2.8, B = Sym (P), ˜p ˜p A where P is a projective A-module of rank 1. Now for each p ∈ Spec(R), P is an p A -module and as A = R [2], we have P is a free A -module since R is a discrete p p p p p p valuationringandhenceextendedfromR . Therefore,byTheorem2.11,P isextended p from R. Let P = P ⊗ A, where P is a projective R-module of rank 1. Hence 1 R 1 B = Sym (P) = Sym (M ⊕Q ⊕P ), where M is a free R-module of rank 1. Since A R 1 1 B = R[3], M ⊕ Q ⊕P is a free R-module of rank 3 [9, Lemma 1.3]. By Theorem 1 1 2.10, Q ⊕P is free of rank 2. Let M = Rf and set S := R[f]. Then B = R[f][2] 1 1 and as f ∈A, we have rank D < 3. Now B = S[2] and D ∈ LND (B) such that D is S fixed-point free. Hence, by Theorem 2.5, D has a slice. Let B = R[f,g,h](= R[3]) and s ∈ B be such that Ds = 1. Then by Theorem 2.1, B(= S[2]) = A[s](= A[1]). Hence by Theorem 2.6, A = S[1]. Let A = R[f,t]. Then B = R[f,g,h] =R[f,t,s] and f,t ∈ A. So rank D = 1. ThefollowingexampleshowsthatTheorem3.6doesnotextendtoahigher-dimensional regular UFD, not even to k[2]. Example 3.10. Let k be a field of characteristic zero and R = k[a,b] = k[2]. Let B = R[X,Y,Z](= R[3]) and D(6= 0)∈ LND (B) be such that R DX = b, DY = −a and DZ = aX +bY. Let u = aX +bY, v = bZ −uX, and w = aZ +uY. Then Du = Dv = Dw = 0, D is irreducible and D2X = D2Y = D2Z = 0. Let A = Ker D. We show that (i) A= R[u,v,w]. 8 (ii) A= R[U,V,W]/(bW −aV −U2), where R[U,V,W]= R[3] and hence A 6= R[2]. (iii) rank D = 3. Proof. (i) Let C := R[u,v,w]. We show that C = A. Clearly C ⊆ A. Note that, B = C [1], so C is algebraically closed in B . But A is algebraically closed in B. a a a a So A = C . Similarly A = C . Since a,b is a regular sequence in C, C ∩C = C. a a b b a b Therefore A ⊆ A ∩A = C ∩C = C. a b a b (ii)Letφ: R[U,V,W](= R[3])։ AbetheR-algebraepimorphismsuchthatφ(U) = u,φ(V) = vandφ(W) = w. Then(bW−aV−U2) ⊆ KerφandbW−aV−U2 isanirre- duciblepolynomialoftheUFDR[U,V,W]. Nowtr.deg (R[U,V,W]/(bW −aV −U2)) = R 2 = tr.deg A. Hence A ∼= R[U,V,W]/(bW −aV − U2). Let F = bW − aV − U2. R Now (∂F, ∂F, ∂F ,F)R[U,V,W] 6= R[U,V,W]. SoAis notaregular ring, in particular, ∂U ∂V ∂W A 6= R[2]. (iii) rank D = 3 by Lemma 3.2. 4 Quasi-nice Derivations In this section we discuss quasi-nice derivations. Let k be a field of characteristic zero, R a k-domain, B := R[n] and m be a positive integer ≤ n. We shall call a quasi-nice R-derivation of B to be m-quasi if, for some coordinate system (T ,T ,...,T ) of B, 1 2 n D2(T ) = 0 for all i ∈ {1,...,m}. Thus for any two positive integers r and m such i that 1 ≤ m < r ≤ n, it is easy to see that an r-quasi derivation is also an m-quasi derivation. We shallcall an m-quasiderivation to bestrictly m-quasi if itis not r-quasi for any positive intger r > m. Over a field K, Z. Wang [14, Theorem 4.7 and Remark 5] has proved the following result for 2-quasi derivations. Theorem 4.1. Let K be a field of characteristic zero and K[X,Y,Z] = K[3]. Let D(=6 0) ∈ LND (K[X,Y,Z]) be such that D is irreducible and D2X = D2Y = 0. K Then one of the following holds: (I) There exists a coordinate system (L ,L ,Z) of K[X,Y,Z], where L and L are 1 2 1 2 linear forms in X and Y such that (i) DL = 0. 1 (ii) DL ∈K[L ]. 2 1 (iii) DZ ∈ K[L ,L ]= K[X,Y]. 1 2 In this case, rank D can be either 1 or 2. (II) There exists a coordinate system (V,X,Y) of K[X,Y,Z], such that DV = 0 and DX,DY ∈ K[V]. In particular, rank D = 1. Conversely if D ∈ Der (K[X,Y,Z]) satisfies (I) or (II), then D ∈ LND (K[X,Y,Z]) K K and D2X = D2Y = 0. The following two examples illustrate the cases rank D = 1 and rank D = 2 for Part (I) of Theorem 4.1. Example 4.2. LetD ∈ LND (K[X,Y,Z])besuchthatDX = DY = 0andDZ = 1. K Then rank D = 1. 9 Example 4.3. Let D ∈ LND (K[X,Y,Z]) such that K DX = 0,DY = X,DZ = Y. Setting R = K[X], wesee thatD ∈ LND (R[Y,Z]) andD is irreducible. By Theorem R 2.2, D = ∂F ∂ − ∂F ∂ for some F ∈ R[Y,Z] = K[X,Y,Z] such that K(X)[Y,Z] = ∂Z ∂Y ∂Y ∂Z K(X)[F][1], gcd (∂F,∂F) = 1. Moreover Ker D = R[F] = R[1]. Setting F = R[Y,Z] ∂Y ∂Z XZ − Y2 we see ∂F = −Y = −DZ and ∂F = X = DY. 2 ∂Y ∂Z Therefore Ker D = K[X,F]. But F is not a coordinate in K[X,Y,Z] since (∂F , ∂F,∂F)K[X,Y,Z] 6= K[X,Y,Z]. So rank D =2. ∂X ∂Y ∂Z We nowaddressQuestion 2of theIntroduction, whichgives apartialgeneralisation of Theorem 4.1. Proposition 4.4. Let R be a PID containing Q with field of fractions K. Let D ∈ LND (R[X,Y,Z]), where R[X,Y,Z] = R[3] such that D is irreducible and D2X = R D2Y = 0. Let D ∈ LND(K[X,Y,Z]) denote the extension of D to K[X,Y,Z]. Let A := Ker D. Suppose D satisfies condition (I) of Theorem 4.1. Then the following hold: (i) rank D < 3. (ii) There exists a coordinate system (L ,L ,Z) of B, such that L ,L are linear 1 2 1 2 forms in X and Y, DL = 0, DL ∈ R[L ] and DZ ∈ R[L ,L ] = R[X,Y]. 1 2 1 1 2 Moreover, A= R[L ,bZ +f(L )], where b ∈ R[L ] and f(L )∈ R[L ,L ]. 1 2 1 2 1 2 Proof. (i) Let (L ,L ,Z) be the coordinate system of K[X,Y,Z] such that D satisfies 1 2 condition (I) of Theorem 4.1. Multiplying by a suitable nonzero constant from R, we canassumeL ∈R[X,Y]. LetL = aX+bY wherea,b ∈ R. Withoutlossofgenerality 1 1 we can assume gcd (a,b) = 1. Since R is a PID, (a,b,0) is a unimodular row in R3 R and hence can be completed to an invertible matrix (say N) in GL (R). Thus L is a 3 1 coordinate in R[X,Y,Z]. As L ∈ KerD = KerD∩R[X,Y,Z], rank D is at most 2 1 and hence rank D ≤ 2 < 3. (ii)NowsetL = L . Sincegcd (a,b) = 1,thereexistc,d ∈ Rsuchthatad−bc = 1. 1 1 R a b 0 X L 1       Hence we can choose N as c d 0 . Then N Y = L . 2 0 0 1 Z Z Now the proof follows from Part (i) of Lemma 2.3. With the notation as above, if D satisfies condition (II) of Theorem 4.1, rank D need not be 1. The following example shows that rank D can even be 3. Example 4.5. Let k be a field of characteristic zero, R = k[t](= k[1]) with field of fractions L and B := R[X,Y,Z](= R[3]). Let D ∈LND (B) be defined by R DX = t, DY = tZ +X2 and DZ = −2X. Then D is irreducible and D2X = D2Y = 0. Let D denote the extension of D to L[X,Y,Z]. Let F = −GX + tY where G = tZ +X2. Then F2 −G3 = tH, where H = tY2−2tX2Z2−2tXYZ−2X3Y −X4Z−t2Z3 ∈ R[X,Y,Z]. SetC := R[F,G,H]. We show that 10