119
In#rza. J. la%(. 5 ,’latin. ScZ.
V,’. 6 N,.
(1983) 119-124
COMMUTATIVITY AND STRUCTURE OF RINGS WITH
COMMUTING NILOPENTS
HAZAR ABU-KHUZAM
Department of Mathematics
University of Petroleum and Minerals
Dahran, Saudi Arabia
ADIL YAQUB
Department of Mathematics
University of California
Santa Barbara, California 93106
(Received September 28, 1981)
ABSTRACT.
Let R be a ring and let N denote the set of nilpotent elements of R.
Let
Z denote the center of R. Suppose that (i) N is commutative, (ii) for every x in R
there exists
x’
<x> such that x- x 2 x
N, where <x> denotes the subring generated
by x, (iii) for every x,y in R, there exists an integer n
(xy)n
(yx)n
and (xy)
n+l
(yx)
n+l
belong to Z.
n(x,y)
1 such that both
Then R is commutative and, in fact,
R is isomorphic to a subdirect sum of nil commutative rings and local commutative
rings.
It is further shown that both conditions in hypothesis (iii) are essential.
The proof uses the structure theory of rings along with some earlier results of the
authors.
KEY WORDS AND PHRASES.
seple ring.
Nil
rng,
@ocal rizg, subdrect sum, ,ubdxrectZ lreducrbe,
1980 )THEMATICS SUBJECT CLASSIFICATION CDE.
1.
Prxmary 16A70.
INTRODUCT ION.
Recently [i], the authors proved that if R is a semisimpIe ring with the property
that, for all x,y in R there exists an integer n
is in the center of
lowing question:
R, then R is commutative.
n(x,y)
such that (xy)
n
(yx)
n
This naturally gives rise to the fol-
what additional conditions are needed to force the commutativity of
H. ABU-KHUZAM AND A. YAQUB
120
R when R is an arbitrary ring?
In this paper, we establish a theorem which gives some
condit ions.
MAIN RESULTS.
2.
In preparation for the proof of the main theorem, we first state the following
result which was proved in [2] (also see [3], [4], [5]):
LEMMA i.
Let R be a
pose that (i) N
the
subrin
ba
rin ..and
let N be the set of nilpotent elements of R.
is commutative, (ii) for every x
<x> generated by x such that x
ab commutes with b.
i__n R there
2
exists an element
N, (iii) for all
x x
Sup-
x’ i__n
R,
a e N and b e
Then R is commutative.
With the aid of the above lemma, we are able to prove the following.
Let R be a ring, N the set of nilpotent elements of R, and Z the
MAIN THEOREMS.
center of
R.
an element
x’
Suppose that (i) N is commutative, (ii) for every x i__n_n R there exists
in the
subrin
every x,y i__9_n R, there
and (xy)
n+l
(yx)
n+l
<x> generated by x such that x
e
N, (iii) for
n(x,y) >_ 1 such that both (xy)
exists an integer n
belong to Z
x2x
n
(yx)
n
Then R is a subdirect sum of local commutative
rings and nil commutative rings.
PROOF.
The proof will be broken into several claims.
CLAIM i.
The idempotents of R are all in the center Z of R.
For, suppose
e
2
e
R, x
R. By hypothesis (iii), there exists a positive
integer n such that
{e(ex
exe
+ e)} n
{(ex
exe
+ e)e} n
This reduces to
(ex- exe + e)
and hence ex- exe commutes with e.
(ex
exe)e
0; that is, ex
above argument, we obtain xe
CLAIM 2.
Therefore, ex
exe.
Replacing ex
e(ex- exe)
exe
exe
+
e by xe
The set N is a commutative ideal in R and hence N
Let a e N, b
By hypothesis (ii),
exe
+
e in the
exe, and Claim I is proved.
This was essentially proved in
proof.
e e’Z
R and let a
k
[4].
O.
2
C Z.
However, for convenience, we re-produce the
121
STRUCTURE OF RINGS WITH COMMUTING NILPOTENTS
(ab)
m
m+l
(ab)
c
for some element c e <ab> and some positive integer m.
Let
(ab)mcm.
Then, as is readily verified,
(ab)me,
(ab)m
Therefore, since e e Z,
2
e
since a
k
e(ab)
e
0.
Thus, e
mcm
aeb(ab)
e
2
ake{b(ab) m-lcm}k
m-I c m
0 and hence by (2.1), (ab)
m
0.
0,
Therefore, ab e N.
Thus ab and ba are in N for all a e N, b e R.
Similarly, ba g N.
(2.1)
(by Claim i).
e g Z
Combining this
with hypothesis (i), we conclude that N is a commutative ideal of R, and hence
If f: R
CLAIM 3.
N2_ Z.
R* is an onto homomorphism then f(N) coincides with the set
of all nilpotent elements of R*.
Again, this was proved in [4], but for convenience we re-produce the proof.
Let d* be an arbitrary nilpotent element of R* with (d*)
that f(d)
d*.
k
0.
Choose d in R such
By hypothesis (ii),
d-
d2d’- e
N for some d’
g
<d>.
(2.2)
Observe that
dk+l(d’) k
d-
(d-
d2d ’)
d2d ’)
+ dd’(d-
+
+
(dd’)k-l(d d2d’).
(2.3)
Since, by Claim 2, N is an ideal in R, the right sde of (2.3) is in N (see (2.2)) and
hen ce
d
Recalling that f(d)
d*, (d*)
d*
and thus d*
CLAIM 4.
k
d
k+l
k
(d
g
(2.4)
N.
O, (2.4) now implies that
f(d)
f(d-
dk+l(d’)k)
f(N),
f(N), which proves Claim 3.
Any homomorphic image
of
R satisfies all the hypotheses (i), (ll),
(iii).
This follows at once in view of Claim 3.
To complete the proof of the Main Theorem, first recall that
R m a subdirect su of rings Ri(i
each R
i
is subdirectly irreducible.
F);
H. ABU-KHUZAM AND A. YAQUB
122
Furthermore, by Claim 4,
(2.6)
each R. satisfies hypotheses (i), (ii), (iii).
1
We now distinguish two cases.
Case i:
R.1 does
not have an identity.
Let x
g
Ro.
1
By hypothesis (ii), there
exists an element y g <x> and a positive integer m such that
m
x
x
m+l
(y
y,
+
m g z ).
<x>,
g
Let
m m
m
m
x e,
e
2
e e
g
(2.7)
<x>).
see (2.1)),
Then, as in the proof of Claim 2
x
(y
x y
e
center of R.; see (2.6) and Claim i).
Z.l
(2.8)
1
Since e is a central idempotent in the subdirectly irreducible ring R i and since R.
m
m
0
0 and, hence by (2.8), x
0. We have thus shown that x
has no identity, e
for all x in
R.I
that is,
Ri;
is nil.
Hence, Ro1 is a nil commutative ring (see (2.6) and hypothesis (i)).
as
5:
R. has
Let x e R..
1
an identity i.
the central idempotent element e in (2.7) and
Arguing as in Case 1
(2.8) satisfies
0
e
If e
0, then x
m
0 (see (2.8)).
or e
On the other hand, if e
m-i m
-i
x
(2.9)
i
=x
y
i, then by (2.7),
e Ri.
We have thus shown that
R.1
Now, let
u be a unit in
exists an integer n
is a local ring (in Case 2)
R.1 and let y e R..
1
(2 10)
By (2.6) and hypothesis (iii), there
n(u,y) > 1 such that
((uy)u
-1)n
(u-l(uy)) n
e Z.
I
center of R.]
1
and hence
n
uy u -i
y
n
commutes with u.
Therefore,
(uynu-i
yn)u
u(uynu-i
Yn),
and thus
STRUCTURE OF RINGS WITH COMMUTING NILPOTENTS
uy
n
uuynu
n
y u
123
yn..
1
Multiplying by u on the right side of the above equation, we get
(uy
Setting y
n- ynu)u
1
+ a,
u(uyn-ynu)
(u
unit in
R.1,
where a is an arbitrary but fixed
(2.11) and recalling that a
k
Z.1 for all k > 2
y E
Ri.
(2. ii)
t_
element of
Ri,
in
(see Claim 4 and Claim 2), we see
that
(u(na)- (na)u)u
u(u(na)
n(ua- au)
(na)u), and thus
(2.12)
commutes with u.
But, by (2.6) and hypothesis (iii), the above argument can be repeated with n + 1
replacing n, to get (see (2.12))
(n + i)(ua- au)
(2.13)
commutes with u.
By (2.12) and (2.13), we conclude that
ua
Moreover, since N.
au commutes with u, (u a unit in R., a e
(2.14)
Ni)._
is commutative (2.14) holds trivially if u is any nilpotent element
of R., and hence by (2.10) and (2.14),
i
ba
ab
commutes with b for all b
a g N
Ri,
R
Therefore, by Lemma i, R.1 is commutative, and thus by (2.10)
i
i.
(2.15)
is a local commuta-
This completes the proof of the Main Theorem.
tive ring.
We conclude with the following.
REMARK i.
Our Main Theorem need not be true if we delete one of the two con-
ditions in hypothesis (iii), as a consideration of the following ring shows:
a
R
b
0
a
0
0
c
2
a
In this ring, we readily verify that (i) N 2
(iii)
(xy)
6
(yx)
6
for all x,y in R (but
Note that R is not commutative.
a, b, c e GF(4)
0
{0},
(xy)7
(ii) x
(yx)7
x
4
e N for all
x e R,
is not in the center of R)
H. ABU-KHUZAM AND A. YAQUB
124
We further remark that we can delete one of the conditions in hypothesis (iii) by
Indeed, the above proof
fixing n and assuming, in addition, that R is n-torsion free.
also shows the following.
THEOREM.
(yx)
n
Le__t
R be a
rin
and let n be a fixed positive integer SUch
e Z and R i__s n-torsion free.
Suppose,
fur.ther
that the
x’ i__n_n
.set
that
(xy)
n
N of nilpotents
<x> such that
2
o_f R
is commutative and for every x
e N.
Then R is a subdirect sum of local commutative rings and nil commutative .rings.
REMARK 2.
i__n_n R
there exists
Our Main Theorem remains valid if we replace the exponent (n
hypothesis (iii) by m, where m is any positive integer relatively prime to n.
the only change in the above proof takes place in (2.13), where (n
+ l)
X
X X
+ i)
in
Indeed,
now gets re-
placed by m, but this does not affect the conclusion in (2.14) or the rest of the
proof.
REFERENCES
i.
ABU-KHUZAM, H. and YAQUB, A. A commutativity theorem for division rings, Bull.
Australian Math. Soc., 2,1 (1980), 43-46.
2.
ABU-KHUZAM, H. and YAQUB, A.
Some conditions for commutatvty of rins with
constraints on nilDotent elements, Math. Japonica, 2__ (1980), 549-551.
3. HIRANO, Y., IKEHATA, S. and TOMINAGA, H. Commutativitv theorems of Outcalt-Yaaub
type, Math. J. Okayama Univ., 21 (1979), Zi-Z.
4. IKEHATA, S. and TOMINAGA, H.
29-30.
5.
PUTCHA. M. and YAQUB, A.
A
comm,,tativitv
theorem, Math. Japonica,
2_4 (1979),
RinRs with constraints on nilpotent elements and commutators, Studia Scientiarum Mathematicarum Hungarica, i2 (1977), 193-197.