Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 463-466
463
A NOTE ON RINGS WITH CERTAIN VARIABLE IDENTITIES
HAZAR ABU-KHUZAM
Department of Mathematical Sciences
King Fahd University of Petroleum & Minerals
Dhahran, Saudi Arabia
(Paper received on June 30, 1987 and in revised form on October 29, 1988)
It
ABSTRACT.
is
proved
constraints of the form [x
m
that
certain
satisfying generalized-commutator
rings
n
n
n
0
with
m and n depending on x and y,
y
y
y
mst have nll commutator ideal.
KEY WDRDS AND PHRASES.
Commutator ideal, periodic ring.
16A70.
1980 AMS SUBJECT CLASSIFICATION CODE.
INTRODUCTION.
1.
xk]
2, let [x I, x2
x
y, denote
2
3
k
A result of Hersteln [I] and of Anan’in and Zyabko [2]
y] by [x,y]
[x, y
k
asserts that if for any x and y in a ring R, there exist positive integers m
m(x,y),
n m
m n
n -n(x,y) such that x y
then the commutator ideal of R is nil. lecently,
y x
Hersteln [3] proved that a ring R in which for any x, y, z R there exists positive
re(x, y, z), n n(x, y, z), and q q(x, y, z) such that
integers m
Let
[x
[[x
m,
Bell
[Xl, x2] denote x
Xk_ I], x k]. For
yn]
[4]
zq]
0
raised
x
2
x
2
xl,
and for k
x and x
x
)
x
must have nil commutator ideal.
the
More recently Klein, Nada and
follwlng conjecture which arises naturally
from
the
above
mentioned work.
Let k
CONJECTURE.
Ix m,
>
I.
If for each x,
y R, there
exists positive integers m and
yn] k
0, then the commutator ideal of R is nil.
In [4], Klein, Nada and Bell proved the conjecture for rings with identity I.
n such that
Given the complexity of
is in sight.
[I] and [3], it would appear that no proof of this conjecture
Our objective is to prove the conjecture for certain classes of rings
and to generalize a result of Hersteln in [3] and some results in [4] and [5].
A ring R is called periodic if for each x in R, there exists distinct positive
n
m
In preparation for the proofs of our main
-x
theorems, we start with the following lemma which is known [5] and we omit its proof.
integers m and n for which x
464
H. ABU-KHUZAM
LEMMA I.
If R is a periodic ring, then for each x in R, there exists a positive
k
integer k--k(x) such that x is idempotent.
2. MAIN RESULTS.
The following theorem shows that the conjecture is true for Artlnian rings
THEOREM I.
Let k
>
I, and let R be an
Artinian ring such that for each x, y in R,
there exists positive integers m and n such that [x,
Y]k
0.
Then the commutator
ideal of R is nil.
PROOF
To prove that the commutator ideal of R is nil it is enough to show that
if R has no nonzero nll ideals then it is commutative.
nonzero nil ideals
So J
Since R is Artinian,
So we suppose that R has no
the Jacobson radical J of R is nilpotent.
0, and hence R is semisimple Artinian.
This implies that R has an identity
element and now, R is commutative by Theorem 3 of [4].
Next, we prove Theorem 2 which shows that the conjecture is true for periodic
This result generalizes a result of Bell in [5].
THEOREM 2.
Let k >
and let R be a periodic ring such that for each x, y in R
rings.
there exists positive integers m and n such that [x
m
yn] k
--O. Then the commutator
ideal of R is nil.
PROOF
If k-- 2, then the result follows by the theorem in [I].
So asse k
>
2
and let x be any element of R and let e be any idempotent of R. By hypothesis, there
m
m
exists integers m and n such that [x
0 This implies that [x
0 and
e] k
hence
m
m
[x
e[x
e
en]k
e]k_l
e]k_l
m
e
Multiplying by e from the right and using the fact that e[x
0 we
m
m
m
m
obtain [x
O. Hence 0
e
([x
e] k-2 e
e[x
)e
[x
e]
e]
e.
k-2
k-2
m
m
m
0 which implies that x e
e]e
Continuing this way we get [x
ex e. Similarily,
m
m
we can get ex
ex e. This implies that
m
m
ex
x e
x
R, e any idempotent and m re(x, e).
(2.1)
e]k_
elk_
Let y be any element of R.
for some positive integer p
Since R is periodic, Lemma
p(y).
implies that
yP
is idempotent
So (2I) implies that for each x, y in R there
exists positive integers m and p such that
xm
yPxm
Now, the result follows by
the well-known theorem in [I] or [2].
THEOREM 3.
Let k
>
I.
If R is a prime ring having a nonzero idempotent element
such that for each x, y in R there exists positive integers m and n such that
yn] k 0. Then R is commutative.
[x m
PROOF. The argument used in Theorem 2 to reach statement (2.1) in the proof shows
m
that a ring satisfying the generalized commutator constraint [x
yn] k 0 must have
its idempotent elements in the center.
For let e and e be idempotent elements in
2
R.
(2.1) implies that e e
e e and hence the idempotents of R commute.
This
2
2
implies that the idempotents of R are central in R [6, Remark 2]. Let e be a nonzero
RINGS WITH CERTAIN VARIABLE IDENTITIES
idempotent of R.
465
Then e is a nonzero central idempotent in the prime ring R.
is an identity element of R since it can not be a zero divisor
Hence e
Now R is eommmtative
by Theorem 3 of [4].
The proof of Theorem 4 below was done by Kezlan in the proof of his main theorem
in
[7]
So we omit its proof here.
THEOREM 4.
>
Let k
I.
If R is a prime ring with a nontrivial center such that
for each x, y in R there exists positive integers m and n such that
[x
m,
yn] k 0, then R is commutative.
The following result generalizes Theorem
of [4].
THEOREM 5.
Let R be a ring and let M be a fixed positive integer
for each x, y E R there exist positive integers m--re(x, y)
M and n
that [x
m,
y
n
PROOF
yn]
belongs to the center of R.
[x
m,
y
we suppose that R has no nil ideals and hence R is a subdirect
Again,
Let Z be the center of R.
yn]
n(x, y) such
Then the commutator ideal of R is nil.
product of prime rings satisfying the above hypothesis of R.
is prime.
n
Suppose that
0 where m
re(x, y)
So we may assume that R
If Z
0, then for each x, y
R,
M, and n n(x, y) This
implies
that
R
is
of [4].
So we may assume that R has a nontrivial center, and
Hence R is comnmtatlve by Theorem 4 above
commutative by Theorem
The following result generalizes Theorem 8 in [3].
z R, there exists positive
q
q(x, y, z) such that [x
y ]n
,z
belongs to the center of R. Then the commutator ideal of R is nil.
PROOF
Again, we may assume that R is a prime ring satisfying the above
hypothesis Let Z be the center of R. If Z
0, then for each x, y, z R,
yn], zq _-0, where m re(x, y, z), n n(x, y, z) and q q(x, y, z). This
[[x
implies that R is commutative by Theorem 8 of [3].
So we may assume that R has a
nontrivial center
For any x, y in R, [x m
Z where m, n, q are each
], yq]
THEOREM 6.
integers m
Let R be a ring in which, for each x, y,
re(x, y, z) n
m,
n(x, y, z) and q
m,
yn
yn yq
m
functions of the variables x and y. So [[ [x
],
], y]
0, which implies
m
ynq] ynq] ynq] 0 Hence R is commutative by Theorem 4 above
that [[ [x
REMARK.
The result in Theorem 6 can be generalized as follows.
R, there exists positive integers m
n
q(x,y,z)
such
that [xm,y
r I, r
2
in R. Then the commutator ideal of R is nil.
such that for each x, y, z
n(x,y,z)
and
elements r
zq
q
r
k
by induction on k and using the argument in Theorem 6.
.....
Let R be a ring
re(x, y, z), n
r
k]
0 for
all
This can be done
We omit the details of the
proof
A(XNOWLEDGEMENT.
Minerals.
This research is supported by King Fahd University of Petroleum and
H. ABU- KHUZAM
466
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6.
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