proceedings of the
american mathematical society
Volume 105, Number 2, February 1989
A LIE PROPERTY IN GROUP RINGS
ANTONIO GIAMBRUNO AND SUDARSHAN K. SEHGAL
(Communicated by Donald S. Passman)
Abstract.
Let A be an additive subgroup of a group ring R over a field
K. Denote by [A,R] the additive subgroup generated by the Lie products
[a,r] = ar-ra,
aeA,
r e R . Inductively, let [A,Rn] = [[A, R„_l],R]. We
prove that [A, Rn] = 0 for some n =►[A, R]R is a nilpotent ideal.
Introduction
Let A be an additive subgroup of a ring R . We define [^4,7?] to be the
additive subgroup of R generated by all Lie products [a ,r] = ar - ra, aeA,
r e R . Inductively, we set
[A,R,R,...,R)
= [A,RH+l] = [[A.RH],R].
n+l
The main result of this note is
Theorem. Let R = KG be the group algebra of a group G over afield K and
A an additive subgroup of R. Then [A , Rn] = 0 for some n => [A , R]R is an
( associative ) nilpotent ideal of R.
Taking A = KG, we conclude that if KG is Lie nilpotent, then
[KG, KG]KG= A(Gf)KG
is nilpotent where A(Cf) is the augmentation ideal of the derived group G1.
This implies that G is abelian if jf(AT),the characteristic of K is 0 and Gf is
a finite p-group if x(K) — P > 0. This is a theorem of Passi, Passman and
Sehgal [3, 5, p. 151]. To prove our theorem we need two results due to Gupta,
Levin and Passman.
Lemma 1 [1]. The nth term Gn of the lower central series of G is contained in
1 + [^■^^^^^^^^^^^
R , R.R]R
-
where R = KG.
n
Received by the editors December 2, 1987 and, in revised form, February 22, 1988.
1980 Mathematics Subject Classification (1985 Revision). Primary 16A27, 16A68.
This research is supported by NSERC, Canada and MPI, Italy.
© 1989 American Mathematical Society
0002-9939/89 $1.00+ $.25 per page
287
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288
ANTONIO GIAMBRUNO AND S. K. SEHGAL
Lemma 2 [4, p. 202]. If KG satisfies a nondegenerate multilinear generalized
polynomial identity then (j>,the FC subgroup of G, is of finite index and has
finite derived group </>'.
In the next section we collect some preliminary results and prove the theorem
in §3.
2. Some preliminary
results
In this section R is an arbitrary ring with 1 and A is an additive subgroup
of R . We denote by CR(A) = {x € R\xa = ax for all aeA}, the centralizer
of A in R . We write [x , y , z] = [[x, y], z] for x, y , z e R .
Lemma 3. (i) For k > 1, [A, Rk]R = R[A , Rk] so that [A, Rk]R is the twosided ideal generated by [A , Rk].
(ii) For a ,b ,x ,y e R we have
[a, by , x] = [a, b , x]y + [a , b][y , x] + b[a , y , x] + [b , x][a, v].
(iii) [R , R , R][A ,R]<[A,R,R]R.
(iv) For b e CR(A), [A , R][b ,R]<[A,R, R]R.
Proof, (i) Let a <=[A , Rk_x], x ,y eR. Then
[a, x]y = (ax - xa)y - a(xy) - (xy)a - x(ay - ya) = [a , xy] - x[a , y]
and
x[a, y] = [a, xy] - [a, x]y.
(ii) is clear.
To obtain (iii) in the expression for [b,x][a,y]
b e[R ,R], x ,y eR to conclude
in (ii), we take aeA,
[R , R , R][A ,R]<[A,R,R]R
+ [A,[R, R]]R + R[A,R,R]
<[A,R, R]R (by the Jacobi identity and (i)).
(iv) We first observe that for a € A , r e R , [a , r]b = [a , rb] and thus
[A , R]b < [A , R]. Now, taking x e[A,R], reR we have
x[b , r] —[xb ,r]-[x
, r]b
which implies
[A , R][b , R] < [[A ,R]b,R] + [A,R,R]R<[A,R,
R]R.
The next lemma is an extension of a result of Jennings [2] to our situation.
The proof parallels that of Jennings.
Lemma 4. Suppose that [A , RJ = 0 for some n > 1. Then [A , R]R is a nil
ideal and [A , R , R]R is a nilpotent ideal.
Proof. We first claim
( *)
lf[A,Rn]
= 0 for some n > 2 then ([A , R„_X]R)2 = 0.
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A LIE PROPERTY IN GROUP RINGS
289
Let u, v e [A , Rn_3] and x, y e R . Then by the Jacobi identity
-[[". x], [v, v]] = [v,[y,[u,
= [[u,x],y
x]]] + [v , [[u, x], v]]
,v]-[[u,x],v
This means that [A,Rn_2] is commutative.
x ,y e R, and use Lemma 3(h) to conclude
,y]e[A,R„]
= 0.
Now, pick a ,b e [A,Rn2],
[a , by , x] = [a , b , x]y + b[a, y , x] + [b , x][a, y].
Since [A,Rn] = 0, we obtain [b ,x][a, v] = 0 i.e. ([A,Rn_x])2 = 0. Due
to the fact that [A ,-R„_,] is central we may conclude that ([A ,RnX]R)2 = 0
establishing our claim (*).
To prove our lemma by induction on n , we may assume that n > 2. First
let n = 2 i.e. [A,R,R] = 0. We have to show that [A, R]R is a nil ideal.
Since [A, R] is central it is enough to prove that [A, R] is nil. In Lemma
3(ii), take aeA,
x e R, x = y, a — b; we get [a, x] =0, proving the
lemma for n = 2. Now, let n > 2. Writing R = R/[A , R„_X]R and A~for the
homomorphic image of A etc., we have [A ,R„_x] = 0. Therefore, due to the
induction hypothesis [A , R]R is nil and [A, R, R]R is nilpotent. Hence due
to (*) we conclude that [A,R]R is nil and [A,R,R]R is nilpotent as desired.
Remark. The statement of our theorem is not true for arbitrary rings. This was
conjectured by Jennings [2] and confirmed by Gupta and Levin [1] by means
of an example. We record another easy example to the same effect.
Example. Let E be the exterior algebra on a countable infinite-dimensional
vector space V over a field of characteristic not 2. Let
V = Spaníüj , v2 , ...} ,
E = Spanfy. vh ■■■vjix
<i2<
< ij.
We have [E,E] = Span{v;|Vj- ■■■v¡ \ir1< i2 < ■■■< i2m} and [E,E,E]
= 0.
It is clear that [E, E] has elements of arbitrarily large index of nilpotency; for
instance, a = vxv2 + v3v4 h-h
v2m_xv2m has index of nilpotency m + 1.
3. Proof
of theorem
In this section we shall fix R = KG, A the given additive subgroup of R ,
f/>the FC-subgroup of G. We shall denote by A(G, N) = (AN)KG the kernel
of the homomorphism KG —>K(G/N) where A(yV) is the augmentation ideal
of the normal subgroup N. We separate a special case as
Lemma 5. Suppose that <f>the FC-subgroup of G is abelian and of finite index.
Then [A , Rn] = 0 for some n=>[A,R] = 0.
Proof. Let m = (G: <j>). Then fix a right transversal {I = xx ,x2, ... ,xm}
of (j>in G. Write (4>,xt) —{a~ix~laxi\a e 4>}and 5(. = A(0,*(.) for i 2, ... , m and S - S2S3 ■Sm. It is well known [5, p. 145, 4 or 3] that there
is an imbedding p: KG —»Mm(K(f>) of KG into m x m matrices over the
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ANTONIO GIAMBRUNO AND S. K. SEHGAL
290
commutative group ring K<f>. Moreover, since Kcj) imbeds in Mm(K(j>) as
scalar matrices,
Mm(K<j>)>K4>-p(KG)>Mm(S).
We may clearly assume that A ^ 0 so that K<f>p(A)^ 0. Since K<f>are scalar
matrices, we have
[K<pp(A),K<j>p(KG).K<pp(KG)] = (K<j>)np([A
,KG,KG, ... ,KG]) = 0.
Let 0/iie
p(A) and write u = J2¡ ¡ xijeu wi*n xn G K-4>,where the e^ are
the usual matrix units. Let i ^ j be fixed; then for sx , ... , sn e S we have
0 = [u,sxeH,s2ejJ,siej},...,snejj]
= J1S2 "Sn
¿Zxiieu-¿Zxiteit'ejj.ejj
= ■■■- ±sxs2 ■■■SfiXyey ± sxs2 ■■•snxj¡ej¡.
Thus S\S2"'snxij = Q an<* since si are arbitrary elements of S which is a
product of augmentation ideals of infinite groups, it follows that x(J = 0 for all
i ^ j. Thus u = Yl'Hi xiieu ■ Suppose that xu ^ 0. Then for j ^ I, we get
0 = [u, sxeu , s2e„.sae„]
= sxs2 ■■■sn[(x„ - xjJ)elj, e„ , ... , e„]
- sxs2- ■■sn(Xa - Xjj)e¡j.
This implies that sxs2---sn(xn - xú = 0 for all s¡ e A((/>,xJ). It follows
that Xu - Xjj for all j — I ,2, ... ,m. Thus u is a scalar matrix. Hence,
0 = [p(A), p(KG)] m p[A , KG] and so [A , KG] = 0.
We are now ready to give the
Proof of the Theorem. We know by Lemma 4 that [A ,R]R is a nil ideal and
[A , R , R]R is a nilpotent ideal. If x{K) = 0, R contains no nonzero nil ideals
[4, p. 47]; so [A , R]R - 0 and we are done. Therefore, we suppose that x(R~) —
p > 0. If G3 is the third term of the lower central series of G then by Lemma
1 we have A(G3) < [R,R,R]R and so it follows that A(G,G3) = A(G¡)R <
[R,R,R]R. Since by Lemma 3(iii), [R,R, R]R[A ,R] = [R,R, R][A , R]R <
[A,R, R]R we get
A(G3)R[A, R] < [A , R , R]R
which is a nilpotent ideal. So A(G3)R[A , R] is a nilpotent ideal of R .
Now, let n: KG —>K(G/G3) be the natural map. If the theorem holds for
nilpotent class two groups, then, we will have
k([A , R]R) = [n(A), n(R)]n(R) is a nilpotent ideal.
So for some m > 1, ([A,R]R)m < A(G3)R and
([A , R]R)m[A , R] < A(C73)R[,4
, R],
a nilpotent ideal.
So ([A , R]R)m+ and hence [A , R]R is a nilpotent ideal, the desired conclusion. Thus we may assume that G is nilpotent of class two, and so, G' is
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291
A LIE PROPERTY IN GROUP RINGS
central. Since [A , Rn] = 0, for aeA,
[a ,xx, ... ,xn] is a multilinear generalized polynomial identity for R . Also, if for all aeA,
[a ,xx]x2 ■•■xn is a
generalized polynomial identity for R , then since 1 e R , we get [A , R] = 0.
The theorem is proved in this case. Thus, we may assume that for some aeA,
[a ,xx, ... ,xn] is a nondegenerate generalized polynomial identity for R . It
follows by Lemma 2 that (C7:<f>)< oo and \<//\ < oo. Let P be the Sylow
/^-subgroup of <f)'. Since A(G,P) is nilpotent, factoring by P, it is enough
to prove the theorem when <f>is a //-group. We use induction on \(f>\. If
\<f>'\=1, </>is abelian and we are done by Lemma 5. Thus we assume \<t>'\> 1.
Since [A , Rn] = 0, then also [A , R „] = 0 where pm is the smallest power of
p greater than n . Thus fox x eR, a e A, we get
n
r
0 = [a , x,x,
i
... ,x] —ax
Pm
-x
Pm
a.
pm
Therefore, for all x e R , xv e CR(A). Now, if for all g e<f>, g
belongs to
the centre of 4>then it follows by Schur's theorem [5, p. 39] that 4> is a p-group
contrary to our assumption that </>'is a nontrivial //-group. Thus there is a
nm
nm
g e (j) such that gv g C(4>), the centre of r/>. Pick h e </>such that g h hgp ¿0. Since gp e CR(A), we have, by Lemma 3(iv), [A,R][gp ,h]<
[A,R, R]R . Thus letting z = h~l g~"mhg"" we get
[A,R]R(l -z) = [A,R]Rh~Xg~pmgpmh(\ -z) = [A , R]R[g"m , h]
= R[A,R][gpm ,h]<[A,R,R]R.
Since the last ideal is nilpotent by Lemma 4, we have that [A ,.R]R(1 - z) is
a nilpotent ideal. Since z is central in G, G being nilpotent of class two, we
have
([A,R]R)k(l-z)k
= 0 forsomeÂ:>l.
Suppose k > 1 and let lR(\- z) be the left annihilator of (1 - z) in R . Then
([A,R]R)k(l -z)k~l
z = (l + z-f--r-z'2'-1).
eR(l-z)r\lR(l
-z) = R(l-z)r\Rz
= 0,
By iterating this process we conclude that
([A,R]R)k(l-z)
= 0.
Factoring G by (z) we conclude by the inductive hypothesis that
(L4,R]R)'<A(C7,(z))
= R(l-z).
Therefore, ([A , R]R)k'(l - z) = 0 and {[A, R]R)kl < R(l - z). It follows that
{[A, R]R)kt <R(l-z)nRz
= 0 as desired.
References
1. N. Gupta and F. Levin, On the Lie ideals of a ring, J. Algebra81 (1983), 225-231.
2. S. A. Jennings, On rings whose associated Lie rings are nilpotent, Bull. Amer. Math. Soc. 53
(1947), 593-597.
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ANTONIO GIAMBRUNOAND S. K. SEHGAL
292
3. I. B. S. Passi, D. S. Passman and S. K. Sehgal, Lie solvable group rings, Cañad. J. Math. 25
(1973), 748-757.
4. D. S. Passman, The algebraic structure of group rings, Wiley, New York, 1977.
5. S. K. Sehgal, Topics in group rings, Marcel Dekker, New York, 1978.
Department
of Mathematics,
University
of Palermo,
Palermo,
Department
of Mathematics,
University
of Alberta,
Edmonton,
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Italy
Canada