Table Of ContentIDEMPOTENTS IN INTERSECTION OF THE KERNEL
AND THE IMAGE OF LOCALLY FINITE
DERIVATIONS AND E-DERIVATIONS
7
1
0
2 WENHUA ZHAO
n
a Abstract. LetK be a fieldofcharacteristiczero,A a K-algebra
J andδ aK-derivationofAorK-E-derivationofA(i.e., δ =IdA−φ
1 for some K-algebra endomorphism φ of A). Motivated by the
2
Idempotent conjecture proposed in [Z4], we first show that for all
] idempotent e lying in both the kernel Aδ and the image Imδ:=
A δ(A) of δ, the principal ideal (e) ⊆ Imδ if δ is a locally finite K-
R derivationoralocallynilpotentK-E-derivationofA;andeA, Ae⊆
. Imδ if δ is a locally finite K-E-derivation of A. Consequently,
h
the IdempotentconjectureholdsforalllocallyfiniteK-derivations
t
a and all locally nilpotent K-E-derivationsof A. We then show that
m
1A ∈Imδ,(ifand)onlyifδissurjective,whichgeneralizesthesame
[ result [GN, W] for locally nilpotent K-derivationsof commutative
1 K-algebrastolocallyfiniteK-derivationsandK-E-derivationsδ of
v all K-algebrasA.
3
9
9
5
0
. 1. Motivations and the Main Results
1
0
Throughout the paper K stands for a field of characteristic zero and
7
1 A for a K-algebra (not necessarily unital or commutative). We denote
v: by 1A or simply 1 the identity element of A, if A is unital, and IA or
i simply I the identity map of A, if A is clear in the context.
X
A K-linear endomorphism η of A is said to be locally nilpotent (LN)
r
a if for each a ∈ A there exists m ≥ 1 such that ηm(a) = 0, and locally
finite (LF) if for each a ∈ A the K-subspace spanned by ηi(a) (i ≥ 0)
is finite dimensional over K.
By a K-derivation D of A we mean a K-linear map D : A → A that
satisfies D(ab) = D(a)b+aD(b) for all a,b ∈ A. By a K-E-derivation
δ of A we mean a K-linear map δ : A → A such that for all a,b ∈ A
Date: January 24, 2017.
2000 Mathematics Subject Classification. 47B47, 08A35, 16W25, 16D99.
Key words and phrases. Locally finite or locally nilpotent derivations and E-
derivations, the image and the kernel of a derivation or E-derivation, idempotents.
TheauthorhasbeenpartiallysupportedbytheSimonsFoundationgrant278638.
1
2 WENHUAZHAO
the following equation holds:
(1.1) δ(ab) = δ(a)b+aδ(b)−δ(a)δ(b).
It is easy to verify that δ is an R-E-derivation of A, if and only if
δ = I−φ for some R-algebra endomorphism φ of A. Therefore an R-
E-derivation is a special so-called (s ,s )-derivation introduced by N.
1 2
Jacobson [J] and also a special semi-derivation introduced by J. Bergen
in [B]. R-E-derivations have also been studied by many others under
somedifferentnamessuchasf-derivationsin[E1,E2]andφ-derivations
in [BFF, BV], etc..
WedenotebyEnd (A)theset ofallK-algebraendomorphisms ofA,
K
Der (A) the set of all K-derivations of A, and Eder (A) the set of all
K K
K-E-derivations of A. Furthermore, for each K-linear endomorphism
η of A we denote by Imη the image of η, i.e., Imη:= η(A), and Ker η
the kernel of η. When η is an R-derivation or R-E-derivation, we also
denote by Aη the kernel of η.
It is conjectured in [Z4] that the image of a LF K-derivation or K-
E-derivation of A possesses an algebraic structure, namely, a Mathieu
subspace. The notion of Mathieu subspaces was introduced in [Z2] and
[Z3], and is also called a Mathieu-Zhao space in the literature (e.g., see
[DEZ, EN, EH], etc.) as first suggested by A. van den Essen [E3].
The introduction of this new notion was mainly motivated by the
study in [M, Z1] of the well-known Jacobian conjecture (see [K, BCW,
E2]). See also [DEZ]. But, a more interesting aspect of the notion is
that it provides a natural generalization of the notion of ideals.
For some other studies on the algebraic structure of the image of a
LF or LN K-derivations or K-E-derivations, see [EWZ], [Z4]–[Z7].
One motivation of this paper is the following so-called Idempotent
conjecture proposed in [Z4], which is a weaker version of the conjecture
mentioned above on the possible Mathieu subspace structure of the
images of LF K-derivations and K-E-derivations.
Conjecture 1.1. Let δ be a LF (locally finite) K-derivation or a LF
K-E-derivation of A and e ∈ Imδ an idempotent of A, i.e., e2 = e.
Then the principal (two-sided) ideal (e) of A generated by e is contained
in Imδ.
Our first main result is the following theorem, which gives a partial
positive answer to Conjecture 1.1 above.
Theorem 1.2. Let K be a field of characteristic zero and A a K-
algebra (not necessarily unital). Then the following statements hold:
1) for every locally finite D ∈ Der (A) and an idempotent e ∈
K
AD ∩ImD, we have (e) ⊆ ImD;
IDEMPOTENTS IN INTERSECTION OF THE KERNEL AND THE IMAGE 3
2) for every locally finite δ ∈ Eder (A) and an idempotent e ∈
K
Aδ ∩Imδ, we have eA, Ae ⊆ Imδ. Furthermore, if δ is locally
nilpotent, we also have (e) ⊆ Imδ.
Note that for every D ∈ Der (A), it can be readily verify that all
K
central idempotents of A lie in AD. Furthermore, by Corollary 2.5 that
will be shown in Section 2, this is also the case for every LN (locally
nilpotent) δ ∈ Eder (A). Therefore we immediately have the following
K
Corollary 1.3. Assume that A is a commutative K-algebra. Then
Conjecture 1.1 holds for all locally finite D ∈ Der (A) and all locally
K
nilpotent δ ∈ Eder (A).
K
Fora different proofof thecorollaryabove forcommutative algebraic
K-algebras, see [Z4, Proposition 3.8]. For a different proof of Theorem
1.2 for algebraic K-algebras (not necessarily commutative), see [Z4,
Corollary 3.9].
Our second main result of this paper is the following
Proposition 1.4. Assume that A is unital and δ is a LF K-derivation
or a LF K-E-derivation of A. Then 1A ∈ Imδ, (if and) only if Imδ =
A, i.e., δ is surjective.
Two remarks about the proposition above are as follows.
First,thepropositionforLNK-derivationsofcommutativeK-algebras
was first proved by P. Gabriel and Y. Nouaz´e [GN] and later re-proved
independently by D. Wright [W]. See also [E2]. During the prepara-
tion of this paper the author was informed that Arno van de Essen
and Andrzej Nowicki have also proved the LF K-derivation case of the
proposition for commutative K-algebras.
Second, if 1A ∈ Aδ, e.g., when δ ∈ DerK(A), the proposition follows
immediately from Theorem 1.2. But, if 1A 6∈ Aδ, the proof needs some
other arguments (See Section 5).
Arrangement: In Section 2 we recall and give a some shorter proof
for van den Essen’s one-to-one correspondence between the set of all
LN K-derivations of A and the set of all LN K-E-derivations of A
(See Theorem 2.1). We also derive some consequences of this impor-
tant theorem that will be needed later in this paper. In Section 3 we
show the K-derivation case of Theorem 1.2. In Section 4 we show the
K-E-derivation case of Theorem 1.2. In Section 5 we give a proof for
Proposition 1.4.
Acknowledgment: The author is very grateful to Professors Arno
van de Essen and Andrzej Nowicki for personal communications.
4 WENHUAZHAO
2. Van den Essen’s One-to-One Correspondence between
Locally Nilpotent Derivations and Locally Nilpotent
E-Derivations
Throughout this section, K stands for a field of characteristic zero,
A for a K-algebra (not necessarily unital or commutative) and I for
the identity map of A.
Denote by D the set of all LN (locally nilpotent) K-derivations of A
and E the set of all LN K-E-derivations of A. We define the following
map:
(2.2) Ξ : D → E
D → I−eD,
where eD:= ∞ Dk.
k=0 k!
With the setting as above we have the following remarkable one-to-
one corresponPdence between D and E, which was first proved by A. van
den Essen in [E1]. See also [E2, Proposition 2.1.3].
Theorem 2.1. The map Ξ : D → E is an one-to-one correspondence
between the sets D and E with the inverse map Ξ−1 given by the fol-
lowing map:
(2.3) Λ : E → D
δ → ln(I−δ),
where ln(I−δ):= − ∞ δk.
k=1 k
For the sake of coPmpleteness, we here give a proof for the theorem
above, which is some shorter than the one given in [E2, Proposition
2.1.3].
First, the following lemma can be easily verified by induction, as
noticed in [E1, E2].
Lemma 2.2. Let B be a ring and δ an E-derivation of B. Then for
all a,b ∈ B and n ≥ 1, we have
n
n
(2.4) δn(ab) = δi(a)δn−i(I−δ)i(b).
i
i=0 (cid:18) (cid:19)
X
Now we can show Theorem 2.1 as follows.
Proof of Theorem 2.1: First, since D is LN, eD is well-defined.
It iswell-known (andalso easy to check directly) thateD isa K-algebra
IDEMPOTENTS IN INTERSECTION OF THE KERNEL AND THE IMAGE 5
automorphism of A. Hence Ξ(D) is a K-E-derivation of A. Consider
∞ Dn
(2.5) Ξ(D) = I−eD = − = Dh(D) = h(D)D,
n!
n=1
X
where
∞ Dn−1 ∞ Dn−2
(2.6) h(D) = −I− = −I−D .
n! n!
n=2 n=2
X X
Since D is LN, and D and h(D) commute, by Eq.(2.5) Ξ(D) is also
LN. Therefore, Ξ is indeed a map from D to E.
Next, we show that Λ(δ) ∈ D for all δ ∈ E. Set D := Λ(δ). Then
δ
∞ δn
(2.7) D = ln(I−δ) = − = δg(δ) = g(δ)δ,
δ
n
n=1
X
where
∞ δn−1 ∞ δn−2
(2.8) g(δ) = − = −I−δ .
n n
n=1 n=2
X X
Since δ is LN, and δ and g(δ) commute, by Eq.(2.7) D is also LN.
δ
Now, let x,y ∈ A. Then by Lemma 2.2 we have
∞
1
(2.9) D (xy) = − δn(xy)
δ
n
n=1
X
∞ n
1 n
= − δi(x)δn−i(1−δ)i(y)
n i
n=1 i=0 (cid:18) (cid:19)
X X
∞
= − δi(x)S (y),
i
i=0
X
where for each i ≥ 0,
(2.10)
∞ ∞
1 n 1 n
S (y) = δn−i(1−δ)i(y) = (1−δ)i δn−i(y).
i
n i n i
n=i (cid:18) (cid:19) n=i (cid:18) (cid:19)
X X
In particular, by Eq.(2.7) and the equation above we have
(2.11) S (y) = −D (y).
0 δ
Claim: S (y) = 1y for all i ≥ 1.
i i
6 WENHUAZHAO
Proof of Claim: For each i ≥ 1 we introduce the formal power series
∞
1 n
(2.12) f (t): = (1−t)i tn−i
i
n i
n=i (cid:18) (cid:19)
X
(1−t)i ∞
= (n−1)(n−2)···(n−i+1)tn−i.
i!
n=i
X
Then S (y) = f (δ)(y). On the other hand, we have the following
i i
identity of formal power series:
(i−1)! di ∞
= − ln(1−t) = (n−1)(n−2)···(n−i+1)tn−i
(1−t)i dti
n=i
X
By Eq.(2.10) and the identity above we have f (t) = 1/i. Hence
i
S (y) = f (δ)(y) = 1y and the claim follows.
i i i
Now by Eqs.(2.7), (2.9), (2.11) and the claim above we have
∞ ∞
D (xy) = − δi(x)S (y) = −xS (y)− δi(x)S (y)
δ i 0 i
i=0 i=1
X X
∞
1
= xD (y)− δi(x)y = xD (y)+D (x)y.
δ δ δ
i
i=1
X
Therefore Λ(δ) = D is a LN K-derivation of A, i.e., Λ is indeed a map
δ
from E to D. Since Ξ and Λ are obviously inverse to each other, we
see that Ξ gives an one-to-one correspondence between D to E, i.e., the
✷
theorem follows.
Next, we derive some consequences of Theorem 2.1. But, we first
need to show the following lemma. Although it is almost trivial, it will
be frequently used throughout the rest of the paper.
Lemma 2.3. Let R be a ring and B an R-algebra. Let F and G be two
commuting R-linear endomorphisms of B such that F is invertible and
G is LN (locally nilpotent). Then F −G is an R-linear automorphism
with the inverse map given by
∞
(F −G)−1 = GkF−k−1.
k=0
X
Proof: Note that F −G = (I−GF−1)F. Since F commutes with
G, so does F−1. Hence U := GF−1 is LN, for G is LN. Therefore the
formal power series ∞ Uk is a well-defined R-linear endomorphism
k=0
P
IDEMPOTENTS IN INTERSECTION OF THE KERNEL AND THE IMAGE 7
of A, which gives the inverse map of I−U. Hence, F−1 ∞ Uk gives
k=0
the inverse map of F −G, from which the lemma follows. ✷
P
Corollary 2.4. Let D, Ξ be as in Theorem 2.1, and D ∈ D. Set
δ = Ξ(D). Then AD = Aδ and ImD = Imδ.
Proof: First, since δ = Ξ(D), we have D = Λ(δ) =:D by Theorem
δ
2.1. Second, by Lemma 2.3 with F = −I and G = D ∞ Dn−2,
n=2 n!
the K-linear map h(D) in Eq.(2.6) is a K-linear automorphism of
A. Therefore, we have AD = Aδ by Eq.(2.5). FurthermoPre, we also
have Imδ ⊆ ImD by Eq.(2.5), and ImD ⊆ Imδ by Eq.(2.7), whence
ImD = Imδ, and the corollary follows. ✷
Corollary 2.5. Let δ be an arbitrary K-derivation of A or a LN K-
E-derivation of A. Then all central idempotents of A lie in Aδ.
Proof: If δ is a K-derivation of D, the corollary actually holds re-
gardless of the characteristic of K, which can be seen as follows. Since
De = De2 = 2eDe, we have (1 − 2e)De = 0. Since (1 − 2e)2 =
1−4e+4e2 = 1, 1−2e is a unit of A. Hence De = 0.
If δ is a LN K-E-derivation of A, then by Theorem 2.1, δ = I−eD
for some LN K-derivation D of A. Since De = 0 as shown above, we
have δe = 0. ✷
3. The Derivation Case of Theorem 1.2
In this section we give a proof of Theorem 1.2 for LF (locally fi-
nite) K-derivations. Throughout this section we let K and A be as
in Theorem 1.2, D a LF K-derivation of A, and e an idempotent in
AD ∩ImD.
Let s ∈ A such that Ds = e. Since De = 0, we have D(ese) =
e(Ds)e = e. So replacing s by ese we assume s ∈ eAe. Furthermore,
0
for convenience we set s = e. Then with the setting above it is easy
to see that for all i,k ≥ 0, we have esi = sie = si and
k(k −1)···(k −i+1)sk−i if i ≤ k
(3.13) Di(sk) =
0 if i > k.
(
We first consider the case that D is LN (locally nilpotent).
8 WENHUAZHAO
Lemma 3.1. Assume that D is LN. For all a ∈ A set
∞ (−1)i
(3.14) φ (a) = Di(a)si
−s
i!
i=0
X
∞ (−1)i
(3.15) ψ (a) = siDi(a).
−s
i!
i=0
X
Then φ (a),ψ (a) ∈ AD and
−s −s
∞
1
(3.16) ae = φ Dj(a) sj
−s
j!
j=0
X (cid:0) (cid:1)
∞
1
(3.17) ea = sjψ Dj(a) .
−s
j!
j=0
X (cid:0) (cid:1)
Note that the case when A is commutative and e = 1 the lemma has
been proven in [GN, W]. See also [E2]. The main idea of the proof
given below is to modify the proof in [GN, W, E2] to the more general
case in the lemma.
Proof: First, by Eqs.(3.13) and (3.14) we have
∞ (−1)i
Dφ (a) = D Di(a)si
−s
i!
i=0
X (cid:0) (cid:1)
∞ (−1)i
= Di+1(a)si +iDi(a)si−1
i!
i=0
X (cid:0) (cid:1)
∞ (−1)j +(−1)j+1
= Dj+1(a)sj = 0.
j!
j=0
X
Hence φ (a) ∈ AD. The proof of ψ (a) ∈ AD is similar.
−s −s
Next we show Eq.(3.16). The proof of Eq.(3.17) is similar.
∞ 1 ∞ 1 ∞ (−1)i
φ Dj(a) sj = Di+j(a)si+j
−s
j! j! i!
j=0 j=0 i=0
X (cid:0) (cid:1) X X
∞
n
= ae+ n! (−1)i Dn(a)sn
i
n=1 i+j=n(cid:18) (cid:19)
X X
i,j≥0
= ae.
✷
IDEMPOTENTS IN INTERSECTION OF THE KERNEL AND THE IMAGE 9
Proposition 3.2. Let δ be a LN (locally nilpotent) K-derivation or
K-E-derivation of A and e ∈ Aδ ∩ Imδ a nonzero idempotent. Let
s ∈ A such that δs = e. Replacing s by ese we assume s ∈ eAe. Set
0
s = e. Then we have
1) if n c si = 0 or n sic = 0 with c ∈ Aδ, then c e = 0 for
i=0 i i=0 i i i
all 0 ≤ i ≤ n. In particular, s is transcendental over the field
P P
Ke;
2) Ae = Aδ[s] and eA = [s]Aδ, where Aδ[s] (resp., [s]Aδ) is the
K-algebra of all polynomials f(s) of the form f(s) = a si
i≥0 i
(resp., f(s) = sia ) with a′s in Aδ;
i≥0 i i P
3) if δ ∈ DerK(AP), then δ |eA= dds and δ |Ae= dds;
4) if δ ∈ EderK(A), then δ |eA= IeA −φ (resp., δ |Ae= IAe −ψ),
where φ (resp., ψ) is the Aδ-algebra endomorphismof eA (resp.,
Ae) that maps s to s+e.
The proof of the K-derivation case of the proposition above is sim-
ilar as the proof of [E2, Proposition 1.3.21]. The K-E-derivation case
follows from Theorem 2.1 and the K-derivation case of the proposition.
So we skip the detailed proof of this proposition here.
From the proposition above we also have the following
Corollary 3.3. Assume further that A is algebraic over K. Let δ be
a LN K-derivation or K-E-derivation of A. Then Aδ ∩Imδ does not
contain any nonzero idempotent of A.
Actually, theLNconditiononδ inthecorollaryabovecanbedropped.
See [Z4, Corollary 3.9]. For more results on the idempotents in the
image of LF or LN K-derivations and K-E-derivations of algebraic K-
algebras, see [Z5] and [Z4].
Next, we consider Theorem 1.2, first, for all LN K-derivations and
K-E-derivations of A.
Lemma 3.4. Theorem 1.2 holds for all LN K-derivations and K-E-
derivations of A.
Proof: Note first that by Theorem 2.1 and Corollary 2.4, it suffices
to show the lemma for all LN D ∈ Der (A). Let e, s and a be as in
K
Lemma 3.1. Then φ (a),ψ (a) ∈ AD for all a ∈ A. By Eq.(3.16) we
−s −s
see that D maps ∞ 1 φ Dj(a) sj+1 to ae, and by Eq.(3.17)
j=0 (j+1)! −s
D maps ∞ 1 sj+1ψ Dj(a) to ea. Hence ea,ae ∈ ImD for all
j=0 (j+1P)! −s (cid:0) (cid:1)
a ∈ A.
P (cid:0) (cid:1)
10 WENHUAZHAO
To show aeb ∈ ImD for all a,b ∈ A, note first that by Eq.(3.16) for
ae and Eq.(3.17) for eb we have
∞
1
aeb = φ Di(a) si+jψ Dj(b) .
−s −s
i!j!
i,j=0
X (cid:0) (cid:1) (cid:0) (cid:1)
ThenD maps ∞ 1 φ Di(a) si+j+1ψ Dj(b) toaeb. Therefore,
i,j=0 i!j! −s i+j+1 −s
we have (e) ⊆ D, i.e., Theorem 1.2, 1) holds for D, as desired. ✷
P (cid:0) (cid:1) (cid:0) (cid:1)
Now we assume that D is LF and consider the case that the base
field K is algebraically closed. In this case D has the Jordan-Chevalley
decomposition D = D +D over K (see [E2, Proposition 1.3.8]) such
n s
that D is semi-simple and D is LN.
s n
Let Λ be the set of all distinct eigenvalues of D and A (λ ∈ Λ)
s λ
the corresponding eigenspace D . Then A has the following direct sum
s
decomposition:
(3.18) A = ⊕ A .
λ∈Λ λ
Actually, the decomposition above gives a K-algebra grading of A,
i.e., A A ⊆ A for all λ,µ ∈ Λ. This is because D and D by [E2,
λ µ λ+µ s n
Proposition 1.3.13] are also K-derivations of A. In particular, A is a
0
K-subalgebra of A. Furthermore, each A is D (and also D and D )
λ s n
invariant. Therefore we have
(3.19) ImD = ⊕ D(A ).
λ∈Λ λ
Lemma 3.5. Assume that K is algebraically closed. Then
1) AD ⊆ A .
0
2) ImD = D (A )⊕ A .
n 0 06=λ∈Λ λ
3) (e) ⊆ ImD for all idempotents e ∈ AD ∩ImD.
L
Proof: 1) By [E2, Proposition 1.3.9, i)] we have
(3.20) AD = Ker D ∩Ker D .
s n
Since Ker D = A , we hence have AD ⊆ A .
s 0 0
2) For each 0 6= λ ∈ Λ, we have D(A ) ⊆ A and
λ λ
D |A = Ds |A +Dn |A = λIA +Dn |A .
λ λ λ λ λ
Since Dn is LN, D |A by Lemma 2.3 is a K-linear automorphism of
λ
Aλ, whence Aλ ⊆ ImD for all 0 6= λ ∈ Λ. Note that D |A0= Dn |A0.
Then by Eq.(3.19) the statement follows.
3) Note that D is a LN K-derivation of A (as pointed out above)
n
and e ∈ Ker D by Eq.(3.20). Applying lemma 3.4 to D we have
n n
(e) ⊆ ImD . Therefore it suffices to show ImD ⊆ ImD.
n n
