On locally nilpotent derivations of Fermat rings
In this note we prove that the ring B²n has non-zero irreducible locally nilpotent derivations, which are explicitly presented, and that its ML-invariant is C.
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irk-123456789-1523052019-06-10T01:26:03Z On locally nilpotent derivations of Fermat rings Brumatti, P. Veloso, M. In this note we prove that the ring B²n has non-zero irreducible locally nilpotent derivations, which are explicitly presented, and that its ML-invariant is C. 2013 Article On locally nilpotent derivations of Fermat rings / P. Brumatti, M. Veloso // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 20–32. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC:14R10, 13N15, 13A50. http://dspace.nbuv.gov.ua/handle/123456789/152305 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this note we prove that the ring B²n has non-zero irreducible locally nilpotent derivations, which are explicitly presented, and that its ML-invariant is C. |
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Brumatti, P. Veloso, M. |
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Brumatti, P. Veloso, M. On locally nilpotent derivations of Fermat rings Algebra and Discrete Mathematics |
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Brumatti, P. Veloso, M. |
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On locally nilpotent derivations of Fermat rings |
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On locally nilpotent derivations of Fermat rings |
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On locally nilpotent derivations of Fermat rings |
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On locally nilpotent derivations of Fermat rings |
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On locally nilpotent derivations of Fermat rings |
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on locally nilpotent derivations of fermat rings |
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Інститут прикладної математики і механіки НАН України |
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2013 |
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On locally nilpotent derivations of Fermat rings / P. Brumatti, M. Veloso // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 20–32. — Бібліогр.: 5 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT brumattip onlocallynilpotentderivationsoffermatrings AT velosom onlocallynilpotentderivationsoffermatrings |
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2025-07-13T02:47:26Z |
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2025-07-13T02:47:26Z |
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1837498212217257984 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 1. pp. 20 – 32
© Journal “Algebra and Discrete Mathematics”
On locally nilpotent derivations of Fermat rings
Paulo Roberto Brumatti and Marcelo Oliveira Veloso
Communicated by V. V. Kirichenko
Abstract. Let Bm
n = C[X1,...,Xn]
(Xm
1
+···+Xm
n
) (Fermat ring), where
m ≥ 2 and n ≥ 3. In a recent paper D. Fiston and S. Maubach
show that for m ≥ n2 − 2n the unique locally nilpotent derivation
of Bm
n is the zero derivation. In this note we prove that the ring
B2
n has non-zero irreducible locally nilpotent derivations, which are
explicitly presented, and that its ML-invariant is C.
Introduction
Let C[X1, . . . , Xn] be the polynomial ring in n variables over complex
numbers C. Define
Bm
n =
C[X1, . . . , Xn]
(Xm
1 + · · · + Xm
n )
,
where m ≥ 2 and n ≥ 3. This ring is known as Fermat ring.
In a recent paper [3] D. Fiston and S. Maubach show that for
m ≥ n2 − 2n the unique locally nilpotent derivation of Bm
n is the zero
derivation. Consequently the following question naturally arises: is the
unique locally nilpotent derivation of the Fermat ring Bm
n for m ≥ 2 and
n ≥ 3 the zero derivation?
In this work we show that the answer to this question is negative for
m = 2 and n ≥ 3. In other words, there exist nontrivial locally nilpotent
derivations over B2
n (see examples 1 and 2). Furthemore, we show that
2010 MSC: 14R10, 13N15, 13A50.
Key words and phrases: Locally Nilpotente Derivations, ML-invariant, Fermat
ring.
P. Brumatti, M. Veloso 21
these derivations are irreducible (see Theorem 2). In the general case,
we prove that for certain classes of derivations of Bm
n the unique locally
nilpotent derivation is the zero derivation (see Proposition 2).
The material is organized as follows. Section 1 provides the basic
definitions, notations and results that are needed in this paper. In section
2 we present some results on the locally nilpotent derivations of the ring of
Fermat. In section 3 we show examples of linear derivations in LND(B2
n)
and some results on these derivations.
1. Generalities
In the following the word "ring" means commutative ring with a unit
element and characteristic zero. Furthermore, we denote the group of
units of a ring A by A∗ and the polynomial ring A[X1, . . . , Xn] by A[n].
A "domain" is an integral domain. If A is a subring of B (A ≤ B) and B
is a domain, then Frac (B) is its field of fractions and trdegA(B) is the
transcendence degree of Frac (B) over Frac (A).
Let R be a ring. An additive mapping D : R → R is said to be a
derivation of R if it satisfies the Leibniz rule: D(ab) = aD(b) + D(a)b, for
all a, b ∈ R. If A ≤ R is a subring and D is a derivation of R satisfying
D(A) = 0, we call D an A-derivation. We denote the set of all derivations
of R by Der(R) and the set of all A-derivations of R by DerA(R). A
derivation D is irreducible if it satisfies: given b ∈ R, D(R) ⊆ bR if and
only if b ∈ R∗.
A derivation D is locally nilpotent if for each r ∈ R there is an integer
n ≥ 0 such that Dn(r) = 0. Let us denote by LND(R) the set of all
locally nilpotent derivations of R. If A is a subring of B, we will make
use of the following notations
LNDA(B) = {D ∈ LND(B) | D ∈ DerA(B)}
KLND(B) = {A; A = ker D, D ∈ LND(B)}.
Given D ∈ LND(B) define νD(b) = min{n ∈ N | Dn+1 = 0}, for
0 6= b ∈ B. In addition, define νD(0) = −∞. The degree function νD
induced by a derivation D is a degree function on B (see [2]).
In this note x, y, z, . . . will represent residue classes of variables X, Y,
Z, . . . module an ideal.
Note that since C is algebraically closed given G =
∑n
i=1 aiX
m
i with
ai ∈ C
∗ there exists a C-automorphism ϕ of C[X1, . . . , Xn] such that
ϕ(Xi) = biXi, bi ∈ C
∗ and ϕ(Xm
1 + · · · + Xm
n ) = G. In this case ϕ
22 On locally nilpotent derivations of Fermat rings
induces a C-isomorphism of the DerC(Bm
n ) in DerC(C[X1,...,Xn]
(G) ). Thus all
the results obtained in this paper about the module DerC(Bm
n ) can be
extended to the module DerC(C[X1,...,Xn]
(G) ). In this paper, derivation of
Fermat ring means C-derivation and therefore we will use the notation
Der(Bm
n ) to denote DerC(Bm
n ).
The following facts are well known (see [1] or [4]).
Lemma 1. Let B be an integral domain and D1, D2 ∈ LND(B) such
that ker D1 = A = ker D2. If there exists s ∈ B such that 0 6= D1(s) ∈ A,
then 0 6= D2(s) ∈ A and D2(s)D1 = D1(s)D2.
Lemma 2. Let B be a domain satisfying ascending chain condition for
principal ideals, let A ∈ KLND(B) and consider the set
S = {D ∈ LNDA(B) | D is an irreducible derivation}.
Then S 6= ∅ and LNDA(B) = {aD | a ∈ A and D ∈ S}.
Proposition 1. Let B be a domain and D ∈ LND(B) a nonzero deriva-
tion. Suppose that A = ker D, then:
a) A is a factorially closed subring of B. In particular B∗ = A∗.
b) If K is any field contained in B then D is a K-derivation.
c) If s ∈ B satisfy Ds = 1 then B = A[s] = A[1].
d) Let S = A \ {0}, then S−1B = (Frac A)[1] and trdegAB = 1.
e) If A′ ∈ KLND(B) and A′ ⊆ A then A′ = A
2. The set LND(Bm
n )
In this section we obtain some results that state that certain classes
of derivations of C[X1, . . . , Xn] do not induce derivations of Bm
n or are
not locally nilpotent if they do.
Let K be a field and let S = K[n]
I
be a finitely generated K-algebra.
Consider the K [n]-submodule DI = {D ∈ DerK(K [n]) | D(I) ⊆ I} of
the module DerK(K [n]). It is well known that the K [n]-homomorfism
ϕ : DI → DerK(S) given by ϕ(D)(g + I) = D(g) + I induces a
K [n]-isomorfism of DI
IDerK(K[n])
in DerK(S). From this fact we obtain
the following result.
Proposition 2. Let d be a derivation of the Bm
n . If d(x1) = a ∈ C and for
each i, 1 < i ≤ n, d(xi) ∈ C[x1, . . . , xi−1] , then d is the zero derivation.
P. Brumatti, M. Veloso 23
Proof. Let F be the Fermat polynomial Xm
1 + · · · + Xm
n . We know
that there exists D ∈ Der(C[n]) such that D(F ) ∈ FC
[n] and that
d(xi) = D(Xi) + FC
[n], ∀i. Thus we have D(X1) − a ∈ FC
[n], and
for each i > 1 there exists Gi = Gi(X1, . . . , Xi−1) ∈ C[X1, . . . , Xi−1] such
that D(Xi) − Gi ∈ FC
[n]. Since D(F ) = m
n∑
i=1
Xm−1
i D(Xi) ∈ FC
[n] and
D(F ) = m
n∑
i=1
Xm−1
i (D(Xi) − Gi) + m
n∑
i=1
Xm−1
i Gi, where G1 = a, we
obtain
n∑
i=1
Xm−1
i Gi ∈ FC
[n] and then obviously Gi = 0 for all i. Thus d
is the zero derivation.
Corollary 1. Let d be a locally nilpotent derivation of the Fermat ring
Bm
n . If d(xi) = αix
m1
1 · · · xmn
n , where αi ∈ C for all i, then d is the zero
derivation.
Proof. Let νd be a degree function induced by a derivation d. Since the
polynomial F is symmetric we can suppose, without loss of generality,
that
νd(x1) ≤ νd(x2) ≤ · · · ≤ νd(xk) ≤ · · · ≤ νd(xn).
Suppose that for some k ∈ {1, . . . , n} we have 0 6= d(xk). Thus
νd(xk) − 1 = m1νd(x1) + m2νd(x2) + · · · + mkνd(xk) + · · · + mnνd(xn).
This implies that mn = mn−1 = · · · = mk = 0. Thus, as d satisfies
the conditions of the Proposition 2, we can conclude that d is the zero
derivation.
3. Linear derivations
This section is dedicated to the study of the locally nilpotent linear
derivation of the Fermat ring.
Definition 1. A derivation d of the ring Bm
n is called linear if
d(xi) =
n∑
j=1
aijxj for i = 1, . . . , n, where aij ∈ C.
The matrix [aij ] is called the associated matrix of the derivation d.
24 On locally nilpotent derivations of Fermat rings
Lemma 3. Let d be a linear derivation of Bm
n and [aij ] its associated
matrix. Then d is locally nilpotent if and only if [aij ] is nilpotent.
Proof. The following equality can be verified by induction over s.
ds(x1)
...
ds(xn)
= [aij ]s
x1
...
xn
. (1)
We know that d is locally nilpotent if and only if there exists r ∈ N such
that dr(xi) = 0 for all i. As {x1, . . . , xn} is linearly independent over C
by the equality 1, we can conclude the result.
Proposition 3. If d ∈ LND(Bm
n ) is linear and m > 2, then d = 0.
Proof. Let A = [aij ] be the associated matrix of d. Thus, for all i, d(xi) =
n∑
j=1
aijxj . Since xm
1 + · · · + xm
n = 0 we infer that
xm−1
1 d(x1) + · · · + xm−1
n d(xn) = 0. Then
0 = xm−1
1 (
n∑
j=1
a1jxj) + xm−1
2 (
n∑
j=1
a2jxj) + · · · + xm−1
n (
n∑
j=1
anjxj)
and as xm
1 = −xm
2 − · · · − xm
n we deduce that
0 = (a22 − a11)xm
2 + · · · + (ann − a11)xm
n +
∑n
j 6=1 a1jxjxm−1
1 +∑n
j 6=2 a2jxjxm−1
2 + · · · +
∑n
j 6=n anjxjxm−1
n . (∗)
Observe that if m > 2, then the set
{xm−1
2 , . . . , xm−1
n }∪{xjxm−1
i ; 1 ≤ i < j ≤ n, }∪{xjxm−1
i ; 1 ≤ j < i ≤ n}
is linearly independent over C. Thus, we can conclude that
a11 = a22 = · · · = ann = a and aij = 0 if i 6= j.
Since d(x1) = ax1 and d is locally nilpotent, we infer that a = 0. Thus,
the matrix A = [aij ] is null and d = 0.
The next result characterizes the linear derivations of the LND(B2
n).
Theorem 1. If d ∈ Der(B2
n) is linear, then d ∈ LND(B2
n) if and only
if its associated matrix is nilpotent and anti-symmetric.
P. Brumatti, M. Veloso 25
Proof. Let d ∈ Der(B2
n) be a linear derivation and A = [aij ] be the
associated matrix of d. Using the same arguments used in Proposition 3
we obtain
0 = (a22 − a11)x2
2 + · · · + (ann − a11)x2
n +
∑
i<j
(aij + aji)xixj
Since the set {x2
2, . . . , x2
n} ∪ {xixj ; 1 ≤ i < j ≤ n} is linearly independent
over C, we know that
a11 = a22 = · · · = ann = a and aij = −aji if i < j,
but if A is nilpotent then its trace na is null and thus A is also anti-
symmetric.
Now we can conclude by Lemma 3 that d is locally nilpotent if and
only if A is nilpotent and anti-symmetric.
The next lemma helps us to find nilpotent and anti-symmetric matri-
ces.
First, we introduce some notation. Given a natural number n > 1,
Mn denotes the ring of matrices n × n with entries in C, In ∈ Mn is the
identity matrix and Sn is the group of permutations of {1, . . . , n}. Let
σ be an element of Sn, Fσ = {i ∈ N; 1 ≤ i ≤ n and σ(i) = i} and
(−1)σ = 1 if σ is even and −1 if σ is odd.
Let A = (aij) ∈ Mn. An elementary result involving A and its charac-
teristic polynomial is given by the following lemma:
Lemma 4. Let A be a matrix in Mn and let
f(X) = det(XIn − A) = Xn + bn−1Xn−1 + · · · + b1X + b0
be the characteristic polynomial of A.
a) If aii = 0 for every i, 1 ≤ i ≤ n, then for all j, 0 ≤ j ≤ n − 1,
bj =
∑
σ∈Fj
(−1)σ(−1)n−j(
∏
i6=σ(i) aiσ(i)), where
Fj = {σ ∈ Sn; ♯(Fσ) = j}. In particular bn−1 = 0.
b) If A is anti-symmetric, then bn−2 =
∑
i<j a2
ij.
Proof. a) Just observe that if C = X.In − A = (cij) and σ ∈ Sn, then
(−1)σc1σ(1) · · · cnσ(n) = (−1)σ(−1)n−♯(Fσ)(
∏
i6=σ(i)
aiσ(i)).X
♯(Fσ).
26 On locally nilpotent derivations of Fermat rings
We know that bn−1 = −trace(A) and then bn−1 = 0 .
b) If σ ∈ Sn then ♯(Fσ) = n − 2 if and only if σ is a transposition,
i.e., σ = (ij), i 6= j. Hence the result is proved as (ij) is odd and
aij = −aji.
Remark 1. Let R be the field of the real numbers. From Theorem 1 and
Lemma 4 we conclude that the zero derivation is the unique derivation of
ring B = R[X1,...,Xn]
(X2
1 +···+X2
n)
that is locally nilpotent and linear.
In the following we present explicit examples of locally nilpotent
derivations of B2
n that are linear.
Example 1. Let n be an odd number and i =
√
−1 ∈ C. Let DI be a
linear derivation of C[n] defined by the anti-symmetric matrix n × n
I =
0 0 . . . 0 0 −1
0 0 . . . 0 0 −i
...
...
. . .
...
...
...
0 0 . . . 0 0 −1
0 0 . . . 0 0 −i
1 i . . . 1 i 0
.
It is easy to verify that
DI(Xn) = X1 + iX2 + · · · + Xn−2 + iXn−1,
and for k < n
DI(Xk) =
{
−Xn, if k is odd.
−iXn, if k is even.
But DI(X2
1 + · · · + X2
n) = 2
∑n−1
i=1 XiDI(Xi) + 2XnDI(Xn) and then
DI(X2
1 + · · · + X2
n) = −2XnDI(Xn) + 2XnDI(Xn) = 0.
Thus, DI induces a linear derivation, dI , of B2
n given by
dI(xn) = x1 + ix2 + · · · + xn−2 + ixn−1,
and for k < n
dI(xk) =
{
−xn, if k is odd.
−ixn, if k is even.
Now is easy to check that I3 = 0. Thus, dI is a locally nilpotent linear
derivation of B2
n by Theorem 1.
P. Brumatti, M. Veloso 27
Example 2. Let n be an even number and let ε be a primitive
(n − 1)-th root of unity . Let DP be a linear derivation of C
[n] defined
by the anti-symmetric matrix n × n
P =
0 0 . . . 0 . . . 0 −1
0 0 0 . . . 0 0 −ε
...
...
. . .
...
. . .
...
...
0 0 . . . 0 . . . 0 −εk
...
...
. . .
...
. . .
...
...
0 0 . . . 0 . . . 0 −εn−2
1 ε . . . εk . . . εn−2 0
It is easy to verify that
DP (Xk) = −εk−1Xn, for k < n
and
DP (Xn) = X1 + εX2 + · · · + εk−1Xk + · · · + εn−2Xn−1.
As in example 1 it is easy to check that DP (X2
1 + · · · + X2
n) = 0. Thus,
DP induces a linear derivation, dP , of B2
n given by
dP (xk) = −εk−1xn, for k < n
and
dP (xn) = x1 + εx2 + · · · + εk−1xk + · · · + εn−2xn−1.
Since 1+ε+ε2 + · · ·+εn−2 = 0 and {1, ε, . . . , εn−2} = {1, ε2, . . . , ε2(n−2)}
it is easy to check that P 3 = 0. Thus, dP is a locally nilpotent linear
derivation of B2
n by Theorem 1.
The next step is to show that the derivations dI and dP are irreducible.
But for this we need the following elementary result.
Lemma 5. Let h be an element of the Bm
n . Then for each k ∈ {1, . . . , n}
there exists a unique G ∈ C[X1, . . . , Xn] satisfying
h = G(x1, . . . , xn) and degXk
(G) < m.
Proof. By the Euclidean algorithm for the ring C[X1, . . . , Xn] it is suf-
ficient to observe that for all k the polinomial F = Xm
1 + · · · + Xm
n is
monic in Xk.
28 On locally nilpotent derivations of Fermat rings
In the Fermat ring B2
n for each k ∈ {1, . . . , n} define the subring Bk of
the ring B2
n by C[x1, . . . , x̂k, . . . , xn] where x̂k signifies that the element
xk was omitted in the ring B2
n .
Lemma 6. Let h ∈ Bn ⊂ B2
n. Then:
1) dP (h) ∈ xnBn if n is even, dP defined in example 2;
2) dI(h) ∈ xnBn if n is odd, dI defined in example 1.
Proof. Suppose that n is even and let h ∈ Bn. Then
h =
∑
i=(i1,...,in−1)
aix
i1
1 · · · x
in−1
n−1 , hence
dP (h) =
∂h
∂x1
dP (x1) + · · · +
∂h
∂xk
dP (xk) + · · · +
∂h
∂xn−1
dP (xn−1)
=
∂h
∂x1
(−xn) + · · · +
∂h
∂xk
(−εk−1xn) + · · · +
∂h
∂xn−1
(−εn−2xn)
then dP (h) ∈ xnBn. The proof of the case n odd is analogous.
Lemma 7. Let h ∈ B2
n. Then
1) dP (h) = 0 if and only if dP (h) = 0 and h ∈ Bn , if n is even;
2) dI(h) = 0 if and only if dI(h) = 0 and h ∈ Bn, if n is odd.
Proof. Suppose that n is even and let h ∈ B2
n. By Lemma 5 there exists
a unique h0, h1 ∈ Bn such that h = h1xn + h0. Assume h1 6= 0. Now note
that
0 = dP (h) = dP (h1)xn + h1dP (xn) + dP (h0). (2)
From Lemma 6 we have dP (h1), dP (h0) ∈ xnBn. Thus, dP (h1) = bxn for
some b ∈ Bn. Hence dP (h1)xn = (bxn)xn = bx2
n = b(−x2
1 − · · · − x2
n−1)
∈ Bn. As dP (xn) = x1 + εx2 + · · · + εi−1xi + · · · + εn−2xn−1 we have
h1dP (xn) ∈ Bn. Thus dP (h1)xn + h1dP (xn) ∈ Bn and by Lemma 6
dP (h0) = cxn for some c ∈ Bn, then by Lemma 5 and (2) we infer that
0 = dP (h1)xn + h1dP (xn) = dP (h1xn). As ker dP is factorially closed
xn ∈ ker dP , so dP (xn) = 0. But since dP (xn) 6= 0, this is a contradiction.
Hence h1 = 0. The proof of the case n odd is analogous.
Lemma 8. Let n ≥ 3 be a natural number. Then
P. Brumatti, M. Veloso 29
1) ker dP = C[x1 − ε(n−2)x2, . . . , x1 − ε(n−k)xk, . . . , x1 − εxn−1], if n
is even.
2) ker dI = C[x1 + ix2, x1 − x3, . . . , x1 − xk−2, x1 − ixk−1], if n is odd.
Proof. Suppose that n is even and let A be the subring
C[x1 − ε(n−2)x2, . . . , x1 − ε(n−k)xk, . . . , x1 − εxn−1]
of Bn
2 . As
dP (x1−ε(n−k)xk) = dP x1−ε(n−k)dP (xk) = −xn−ε(n−k)(−ε(k−1)xn) = 0,
for every k < n, we deduce that A ⊆ ker dP . Given
y2 = x1 − ε(n−2)x2, . . . , yk = x1 − ε(n−k)xk, . . . , yn−1 = x1 − εxn−1
observe that
A[x1] = C[x1, y2, . . . , yn−1] = C[x1, . . . , xn−1] = Bn,
thus the set {x1, y2, · · · , yn−1} is algebraically independent over C.
By Lemma 7 for each h ∈ ker dP , we have dP (h) = 0 and h ∈ Bn, then
we may write h =
n∑
i=0
aix
i
1 where ai ∈ A ⊆ ker dP for all i ∈ {0, . . . , n}.
Assume n > 0 and remember that dP (x1) = −xn. So
0 = dP (h) = −[a1 + 2a2x1 + · · · + nanxn−1
1 ]xn.
By the uniqueness of Lemma 5 we have a1 + 2a2x1 + · · · + nanxn−1
1 = 0
and hence ai = 0 for i = 1, . . . , n. Therefore h = a0 ∈ A ⊆ ker dP . The
proof of the case n odd is analogous..
Theorem 2. Let n ≥ 3 be a natural number.
1) If n is even, then dP ∈ LND(B2
n), where dP was defined in the
example 2, is irreducible and
LNDA(B2
n) = {adP | a ∈ A},
where A = C[x1 − ε(n−2)x2, . . . , x1 − ε(n−k)xk, . . . , x1 − εxn−1].
2) If n is odd, then dI ∈ LND(B2
n), where dI was defined in the
example 1, is irreducible and
LNDS(B2
n) = {sdI | s ∈ S},
where S = C[x1 + ix2, x1 − x3, . . . , x1 − xn−2, x1 − ixn−1].
30 On locally nilpotent derivations of Fermat rings
Proof. Suppose that n is even and d ∈ LNDA(B2
n)\{0}. By Proposition 1
we have ker d = A. Observe that
d2
P (xn) = dP (
n−1∑
k=1
εk−1xk) =
n−1∑
k=1
εk−1dP (xk) = xn(
n−1∑
k=1
ε2(k−1)) = 0
thus dp(xn) ∈ A. Then, by Lemma 1, d(xn) ∈ A and
dP (xn)d = d(xn)dP . (3)
By definition dP (x1) = −xn, so
dP (xn)d(x1) = −d(xn)xn. (4)
We know that d(x1) = g1xn + g0 with g0, g1 ∈ Bn. Then, (4) implies that
dP (xn)g1xn + dP (xn)g0 = −d(xn)xn. Since dP (xn) ∈ A ⊆ Bn, by the
uniqueness of Lemma 5 we obtain d(xn) = −dP (xn)g1. As d(xn) ∈ A we
know that dP (d(xn)) = 0. Thus 0 = dP (d(xn)) = dP (−dP (xn)g1) and
then dP (g1) = 0, i.e., g1 ∈ A. Since d(xn) = −dP (xn)g1, (3) implies that
dP (xn)d = d(xn)dP = −dP (xn)g1dP .
Therefore d = −g1dP , where −g1 ∈ A. The Lemma 2 implies that
dP = hd0 for some h ∈ A and some irreducible d0 ∈ LND(B2
n). As
we saw d0 = h0dP for some h0 ∈ A. So dP = hd0 = h(h0dP ) = (hh0)dP .
Thus h ∈ A∗ = C and hence dP is irreducible. The proof of the case n
odd is analogous.
Let B be a C-domain and θ ∈ AutC(B). It is well known that if
D ∈ LND(B), then θDθ−1 ∈ LND(B) and ker θDθ−1 = θ(ker D).
Let Sn be the symmetric group and σ ∈ Sn. The permutation σ
induces a C-automorphism of C[n] = C[X1, . . . , Xn] which is also called σ
and defined by relations σ(Xi) = Xσ(i) for every i. Now since
σ(X2
1 + · · · + X2
n) = X2
1 + · · · + X2
n
then σ induces a C-automorphism of B2
n which is also called σ and defined
by relations σ(xi) = xσ(i) for every i. Suppose that n is even. Given j < n
we denote the transposition (j n) ∈ Sn by τj and the derivation τjdP τj
−1
by dPj
. Hence we have dPj
∈ LND(B2
n) and
ker dPj
= τj(C[x1 − ε(n−2)x2, . . . , x1 − ε(n−k)xk, . . . , x1 − εxn−1]).
P. Brumatti, M. Veloso 31
Observe that
τj(x1 − ε(n−k)xk) =
xn − ε(n−k)xk, if j = 1
x1 − ε(n−k)xn, if j = k
x1 − ε(n−k)xk, otherwise.
This implies that ker dPj
⊂ Bj .
Now suppose that n is odd. For each 1 ≤ j ≤ n denote the derivation
τjdIτj
−1 by dIj
. Thus we have
ker dIj
= τj(C[x1 + ix2, x1 − x3, . . . , x1 − xn−2, x1 − ixn−1]).
if k is odd
τj(x1 − xk) =
xn − xk, if j = 1
x1 − xn, if j = k
x1 − xk, otherwise.
If k is even
τj(x1 − ixk) =
xn − ixk, if j = 1
x1 − ixn, if j = k
x1 − ixk, otherwise.
Is follows that ker dIj
⊂ Bj .
The concept of ML-invariant of the a ring R was introduced by
L. Makar-Limanov. This invariant has proved very useful in studying the
group of automorphisms of a ring (see [5]) .
Definition 2. Let B be a ring. The intersection of the kernels of all
locally nilpotent derivation of B is called the ML-invariant of B.
The next result shows that the ML-invariant of B2
n is C. Note that
for m ≥ n2 − 2n the ML-invariant of Bm
n is Bm
n .
Theorem 3. The ML-invariant of the ring B2
n is C.
Proof. We define dj = dIj
if n is odd, and dj = dPj
if n is even. In both
cases, by previous observations, we have ker dj ⊂ Bj and
∩n
j=1ker dj ⊂ ∩n
j=1Bj = C.
Since C ⊂ ker dj , for all j ∈ {1, . . . , n}, then the result follows.
References
32 On locally nilpotent derivations of Fermat rings
[1] D. Daigle, Locally nilpotent derivations, Lecture notes for the Setember School of
algebraic geometry, Lukȩcin, Poland, Setember 2003, Avaible at
http://aix1.uottawa.ca/~ddaigle.
[2] M. Ferreiro, Y. Lequain, A. Nowicki, A note on locally nilpotent derivations, J.
Pure Appl. Algebra N.79, 1992, pp.45-50.
[3] D. Fiston, S. Maubach, Constructing (almost) rigid rings and a UFD having
infinitely generated Derksen and Makar-Limanov invariant, Canad. Math. Bull.
Vol.53 N.1, 2010, pp.77-86.
[4] G. Freudenberg, Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia
of Mathematical Sciences, Vol.136, Springer-Verlag Berlin Heidelberg 2006.
[5] L. Makar-Limanov, On the group of automorphisms of a surface xny = P (z),
Israel J. Math. N.121, 2001, pp.113-123.
Contact information
P. Brumatti IMECC-Unicamp, Rua Sérgio Buarque de
Holanda 651, Cx. P. 6065
13083-859, Campinas-SP, Brazil
E-Mail: brumatti@ime.unicamp.br
M. Veloso Defim-UFSJ, Rodovia MG 443 Km 7
36420-000, Ouro Branco-MG, Brazil
E-Mail: veloso@ufsj.edu.br
Received by the editors: 06.09.2010
and in final form 05.04.2013.
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