Internat. J. Math. & Math. Sci.
(1985) 145-149
145
Vol 8 No.
ON CENTER-LIKE ELEMENTS IN RINGS
JOE W. FISHER and MOHAMED H. FAHMY
University of Cincinnati
45221
Cincinnati, Ohio
U.S.A.
4, 1982 and
(Received June
I, 1982)
in revised form Septmber
In a paper with a similar title Herstein has considered the structure of
0 or is in the
prime rings which contain an element a which satisfies either [a x in
ABSTRACT.
R
center of
for each
x
We consider the structure of rings which contain an
R.
in
The two types which
element a which satisfies powers of certain higher commutators.
we consider are
.,x m
Xl,X 2
R.
in
[[a,xl],X2] .,Xm ]n 0
[a,[xl,[X2, ..,[Xm_ I
(i)
R
in
R
or is in the center of
and (2)
x
]]]n
m
0
We obtain results similar to those obtained by Herstein;
for all
x
for all
Xl,X 2
however,
in some
m
cases we must strengthen the hypotheses
Also we consider a third type (3) (ax
n
m
x a)
k
for all
0
x
in
R
and extend
results of Herstein and Giambruno.
KEY WORDS AND PHRASES. Prime and semiprime rings, primitive and semiprimitive rings,
rings with involution, commutativity theorems.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 16A12, 16A20, 16A28, 16A70
INTRODUCTION.
i.
The definition of the center
several papers.
R
is central if
(I) if
a e Z.
identity
R
has recently been generalized in
R
of a ring
an element
a
of a prime ring
e U
U
is a nonzero two
where
We generalize this result in two directions.
R.
sided ideal in
Z
[i, Theorem 2] showed that
0 for all u
and only if [a,u in
Herstein
[[a,ul],U 2]
is prime and
.,Um ]n
0
First, we show that
for all
U,
g
then
Ul,U2,...,u m
From (i) it follows easily that a semiprime ring satisfying the Lie nilpotent
[...[Xl,X2],...
0
x
m
for all x
I x2,...,xm
in
R
is commutative
p. 230].
We also conclude from (i) two commutivity theorems which generalize two
well-known theorems due to Kaplansky
2, p. 219] and Herstein [3].
2,
I
R
u
u
2
m
is prime with
Herstein
all
in
x
U, then
Z
a
g
Z
infinite and
if either
n
R
is semisimple and
R
0
U
for all
is essential, or
fixed.
[i, Theorem 4] proves that if
R, then
[Um-l’Um]’’’]]]n
[a’[ul’[Ul’[U2
Second, we prove that if
u
R
is prime, a
Z, and [a,x ]n
is an order in a 4-dimensional simple algebra
g
Z
for
We show that the
J. W. FISHER AND M. H. FAHMY
146
,Xm in Z for all Xl,X 2
In another attempt to generalize the structure of the center
[...[[a,xl],X 2]
same result holds if
,Xm
Z
R.
e
R
of a ring
without nonzero nil ideas Herstein [4] proved that the subring
Z. This theorem was generalized
for all x e R}
{a e R: axn(a’x)
x
T
n(a’X)a
{a e R:
by Giambruno
[5],
x e R}
Z.
In an attempt to generalize these
{a
R:
(axm(a
G
a)
k
xn(a’X)a
m(a’x)
results, we show that
Z if R
for all x in R}
0
R.
0
rx
Moreover, [a,x]
is an associative ring with 1 and
R
Throughout this paper
2.
x)
xn(a
x)
ax
for all
is semiprimitive
2R # 0.
and
of
G
who showed that the set
denotes the center
{r e R:
(X)
R, then
is a subset of
X
and if
xa
ax
Z
x e X}.
for all
We begin this section with a lemma which will be useful in the sequel.
0
Let R be a ring, U an ideal of R, and a e R. If
2
0 for all u e U.
then [a,u]
e U,
MAIN RESULTS.
[[a,ul],U 2]
LEMMA i.
for all
ul,u 2
PROOF.
Let
U.
u e
However the first term is zero.
Let
THEOREM i.
Hence
Ul,U2,...,um
PROOF
a[[a,u],u]+ [a,u
[a,u]
2
0.
U # 0
be a prime ring and
R
that for fixed positive integers
e
is an ideal we obtain
=[a[a,u],u]
[[a,au],u]
0
U
Since
n, [.. [[a,u I ]u 2
and
m
R.
If
],...,urn ]n
0
an ideal of
a e R
is such
for all
a e Z.
U, then
The proof goes by induction on
m.
The result is true for
m
i
by
Herstein’s theorem [i, Theorem 2].
Assume the result is true for
for all
e U.
Ul’U2’’’’’Um
Set
k
m
U.
u2,u3,...,Um
Lemma i we obtain that
[a,u]
b e Z
Hence
2
0
By applying
by the induction hypothesis.
for all
0
0
Then by assumption
.,Um ]n
[[b,u 2],u 3],
for all
[[a’ul]"12]’’’’’Um in
and suppose that
[a,u ].
b
u e
U.
Therefore
a e Z
by Herstein’s
aforementioned theorem.
As a consequence of Theorem i, we get the following two corollaries which generalize
for prime rings two well-known theorems due to Kaplansky [2
p. 219] and Herstein [3].
COROLLARY i.
a
R
Let
R
[.. [[a k(a) u
where
Ul,U2,...,Um(a)
PROOF.
R
I] u2],.
R
Um(a) ]n(a)
0
is commutative.
Evident.
COROLLARY 2.
a
U, then
e
U # 0 an ideal of R. If for every
k(a), m(a), and n(a) such that
be a prime ring and
there exists three natural numbers
Let
R
be a prime ring and
there exists two natural numbers
U # 0
re(a), n(a) and
and ideal of
R.
a polynomial
Pa ()
coefficients such that
[...[[(a
where
Ul,U2,...,Um(a)
a
2
Pa(a),ul] u 2]
U, then
R
Um(a) ]n(a)
is commutative.
0
If for every
with integer
CENTER-LIKE ELEMENTS IN RINGS
PROOF.
147
Evident.
2, p. 230].
Also as a corollary we obtain a result from
COROLLARY 3.
R
If
[...[x l,x 2],...,x
n]
is a semiprime ring satisfying the Lie nilpotent identity
R
0, then
is commutative.
Ev iden t.
PROOF.
[i, Theorem 3].
Z and let a e R,
The next theorem generalizes a theorem of Herstein
R
Let
THEOREM 2.
be a prime ring with center
Um ]n
[...[[a,ul],U 2]
such that
R
e Z
for all
R.
Then
PROOF.
If
[...[a,ul],U2],...,Um_ I]
Z
e Z
Hence there exists
by Theorem 2.
for all
[b,um in
e Z
Vm_lJ
for all
u
m
e
urn_ I
Ul,U 2
U
Vl,V2,...,Vm_ I
[...[[a,vl],V 2]
b
However by hypothesis
Ul,U 2,...,urn
Z
#
U
where
be
0
is
is an order in a 4-dimensional simple algebra.
an ideal of
a
e U
a
U,
e
then
such that
Z.
U.
Ergo,
R
is an order in a
4-dimensional simple algebra by Herstein [i, Theorem 3].
We now generalize Herstein’s Theorem 2 in [i] in another direction.
R, a
be an ideal of
R, m
e
[a,[ul,[U 2
for all
Ul’U2’’’’’Um
+.
Z
fixed in
U # 0
Let
If
[Um_ I
u
]]in
m
0
e U (Condition A) then we shall prove
a e Z
that
in the
following two cases:
(i)
R
is semiprimitive and
(ii)
R
is prime,
U
is an
U
(U)
is an ideal such that
Z
ideal,
is infinite, and
n
0
(Theorem 3), or
is fixed. (Theorem 4).
First we prove a lemma:
LEMMA 2.
Let
R
fying condition (A).
(a)
PROOF.
for all
x
I x2,..,
(b)
be a primitive ring,
Then
R
If
m
If
U # 0
R
an ideal of
Hence
Z
a
[al[Xl,[X2,... [Xm_ I
V
R
a
satis-
x
.J]]
m
0
by a result of Smiley [6
is primitive, then it has a faithful irreducible R-module
which is also faithful and irreductible as a U-module.
densely on
R, and
Z.
is a division ring, then
R.
x
a e
as a vector space over a division ring
and the result follows from (a).
So let
dimDV
V
By the Density theorem U acts
D.
If
dimDV
i, then
R
D
i.
Suppose that there exists a nonzero victor v e V such that v and va are
linearly independent over D. Since U acts densely on V there exists
Ul,U 2 e U
such that vu
(va)u
vu
and
(va)u
v,
v,
Thus
0,
va.
I
I
2
2
and so
v[a,[ul,[Ul,[Ul,...,[Ul,U2]...]]]
v[a,[ul,[Ul,. .,[Ul,U 2] ]]]n v. But, by the
the left is zero, which gives that
v e
V,
va
%(v)v, where
does not depend on
and (va)x
=’(%v)x
faithfully on
V
we have
ax
xa
hypothesis, the expression on
0, contrary to our assumption.
v
%(v) e D.
v, hence va
%(vx). Hence
v
Thus for every
It follows easily from this that, in fact,
%v
for all
v(xa
0
ax)
for all
v e V.
0
So, if
for all
v e V.
x e R, and so
(v)
x e R, then (vx)a
Since
a e 7.
R
acts
%vx
J. W. FISHER AND M. H. FAHMY
148
THEOREM 3.
a eR
and
shown that
product of
R/P
R/P.
R
Z
U.
P
where
a
Let
R
UC.
If
#
a
Z
then
A,
and
R [7 ].
Z
C
Then
RC
S
and because
and its ideal
carries over to the prime ring
R
satisfies a nontrivial generalized polynomial identity
we have that
I:
R,
on ideal of
A
0
primitive by Martindale’s theorem.
Question
is the subdirect
a e Z.
A, then
[a’[ulx’[u2x’’’’’rUm-lX’UmX]’’’]]]n
condition
it can easily be
is in the center of
a
U # 0
infinite,
be the extended centroid of
C
is infinite condition
V
Z
be prime with
which satisfies condition
Let
Lemma 2 that
It follows from
O,
Z.
Z.
Therefore
PROOF.
a e
0, U is an essential ideal of R. Hence
U}
O. Hence R
P primitive ideal such that P
THEOREM 4.
a
A, then
with (U)
R
an ideal of
0
E(U)
Since
n{P:
U
is a semiprimitive ring,
which satisfies condition
PROOF.
each
R
If
Z(S)
a e
for
Um
Ul,U 2
V
Since
UC
by Lemma 2.
R.
e
S
is
which satisfies
a e Z.
Hence
In Theorem 4 is the hypothesis that
RC
S
Hence
is an ideal of
be infinite necessary?
Z
Note
that in Theorem 1 it was not necessary.
We finish our paper with a partial generalization of the results in [5] and [4].
Let a be an element of the ring
such that for all
R
u e
U,
R,
a nonzero ideal of
we have
n(u)
(aum(U)
k(u)
(Condition B)
0
xn(X)a)k(x)
{a e R: (axm(x)
and let
a)
0
for all
x
R}.
in
It is clear that
-G mT =Z.
THEOREM 5.
(i)
Then
R
If
is a ring satisfying condition
E(U)
(B) with
2R # O, then either
R
is semiprimitive with
(2)
R
is prime with infinite center with fixed integers m,
a e
Z.
0
or
n, and k.
By using the same technique of proof as that in Theorems 3 and 4, it is
PROOF.
enough to prove the result in the primitive case.
Let
V
be as in the proof of Lemma 2.
D
and
division ring, we get that for all
Giambruno [8] a e Z.
Thus let
x e
dimDV
R,
I.
m(x)
ax
If
0
If
xn(X)a
v e
I, i.e., R
dimDV
O.
is a
Hence by a result of
V, then the vectors {v,va,va
2
are linearly dependent.
Indeed, if they were linearly independent, then by the density
such that vu
O.
v, (va)u
v, and (va2)u
m(u)
n (u) k (u)
Thus we get v(au
v if k(u)
a
va if
u
is even and equals v
m(u)
is odd.
But (au
0 so we get a contradiction in both cases.
u e U
theorem, there is
K(u)
un(U)a)k(u)
{v,va}
Assume that
%,
and
e D.
%
If
(va)w
However,
(va)y
m(y)
v(ay
@ 0,
So
0.
and
if
yn
Contradiction.
%
0, i.e.,
va
#
D,
0
a)
Therefore
v
e
2
y(va)
{v,va}
the proof of Lemma 2 shows that
RE,LARKS: i)
va
then by the density theorem there is w
n(w) k(w)
s
m(w)
v(aw
+% v where s
a)
-w
where
(Y) k (Y)
av
are linearly independent, then
va,
then there is
(because
a
0
where
0
2R
B
are linearly dependent.
a
Z.
y
2
+ va where
%v
U
such that
s(k).
U
such that
0
implies
B(k)
D
and
vw
v
Contradiction.
D #
y
vy
2y(k)
v
).
Thus
e
D.
The same argument as used in
This completes the proof.
It is of interest to study all the above theorems for rings with
CENTER-LIKE ELEMENTS IN RINGS
"*"
involution
by applying the same conditions on the set of symmetric elements
For example, it is natural to ask:
and
Z, then what about
a
[Sl,S2,..
2)
s
n
in
149
e
Z,
R?
thn dimzR
If
[a,sl,...,sn in
e Z
for all
Sl,S2,...,s n
[8], that
It was shown by Fahmy [9] and Giambruno
<_
S
if
16.
A second direction in which one may try to extend the above theorems is to
generalize the cohypercenter introduced by Chacron in 0], i.e., to study the set
2
{a
is a polynomial with
R: [a,x
x p(x)
0 for all x in R} where p(x)
in(x)
integral coefficients.
REFERENCES
I.
HERSTEIN, I. N. Center like elements in prime rings, J. Algebra 60, (1979), 567-574.
2.
JACOBSON, N. Structure of rings, Amer. Math. Colloquim Publications, V 37 (1964).
3.
HERSTEIN, I. N. Structure of a certain class of rings, Amer. J. Math 75, (1953),
864-871.
4.
HERSTEIN, I. N. On the hypercenter of a ring, J. Algebra 36, (1975), 151-157.
5.
GIAMBRUNO, A. Some generalization of a center of a ring, Rend. Circ. Mat. Palermo
(2), 27 (1978), 270-282.
6.
SMILEY, M. F. Remarks on the commutivity of rings, Proc. A.M.S. IO (1959), 466-470.
7.
MARTINDALE, W. S. Prime rings satisfying generalized polynomial indentity,
Alsebra 12, (1969), 576-584.
8.
GIAMBRUNO, A. Algebraic conditions for rings with involution, J. Algebra 50, (1978),
190-212.
9.
FAHMY, M. H. Rings with involution and Polynomial identity, Ph.D. dissertation,
Moscow, 1976, (Russian).
!0.
CHACRON, M. A Commutativity theorem for rings, Prco. Amer. Math. Soc. 59 (1976),
No. 2, 211-216.
J_.
11.
HERSTEIN, I. N. Rings with involution, The University of Chicago Press (1976).
12.
HERSTEIN, I. N., PROCESI C., and SCHACHER, M., Algebraic valued functions on
noncommutative rings, J. Alsebra 36, (1975), 128-150.