KOMATSU, HIROAKI

Osaka City University Repository

Komatsu, H.
Osaka J. Math.
22 (1985), 811-814
A COMMUTATIVITY THEOREM FOR RINGS. II
HIROAKI KOMATSU
(Received October 22, 1984)
Throughout the present paper, R will represent a ring with center C, and
D the commutator ideal of R. A ring R is called left (resp. right) ί-unital if
x^Rx (resp. x&xR) for every x^R; R is called s-unital if R is both left and
right s-unital. Given a positive integer ny we say that R has the property Q(n)
if for any x, y^R, n[x, j]=0 implies [x, y]=0 (see [1]).
Our present objective is to generalize [2, Theorem] for left ί-unital rings
as follows:
Theorem. Let n>0, r, s and t be non-negative integers and let f(Xy Y)=
r s
Σ Σ fij(X> Y) be a polynomial in two noncommuting indeterminates X, Y with
1=1 j=2
integer coefficients such that each fa is a homogeneous polynomial with degree i
in X and degree j in Y and the sum of the coefficients of ffj equals zero. Suppose
a left s-unital ring R satisfies the polynomial identity
(1) X'[Xn, Y]-f(X, Y) = 0.
If either n=l or r=l and R has the property Q(ri), then R is commutative.
We shall use freely the following well known result stated without proof.
Lemma. Let x, y be elements of a ring with 1, and let k be a positive integer.
If^y=0=(x+l)ky theny=0.
Proof of Theorem. Let y be an arbitrary element of Λ, and choose an
element e of R such that ey=y. Then (1) gives y—yen—f(e, y)^yR. We
have thus seen that R is right s-unital, and hence s-unital. Therefore, in view
of [1, Proposition 1], it suffices to prove the theorem for R with 1.
Observe that D is a nil ideal of R, by a theorem of Kezlan-Bell (see, e.g.,
[1, Proposition 2]), since x=e
u
 andy=e12 fail to satisfy (1).
I) We consider first the case n— 1. Let αy b be elements of R. By
Lemma, it is easy to see that if x*α\xy b]=Q for all x^R then α[xy b]=0. Not-
ing this fact, we can apply the argument employed in the proof of [2, Theorem]
to see the commutativity of R.
812 H. KOMATSU
II) Next, suppose that n>l, r=l and R has the property Q(n). We
claim that Dc C. In fact, if a^D\C then there exists a positive integer p
such that ap^C and ak^C for all k>p. For any jeT?, by repeated use of
(1), we have n(l+ap)*[ap, y]=(l+ap)*[(l+ap)*, y]=f(ί+ap, y)=f(^y y)+f(apy y)
=f(ap, y)=apt[apn, y]=0. Since l+ap is a unit in R, we have
Hence, [ap, y]=0 by β(w), a contradiction. We have thus seen that D^C.
We write f
υ
(X, Y)=Σ «y* -^ί^ "*. Since Σ α, *=0 by assumption, we have
. , *=o . *=o .
Therefore, we can write f(X, Y)=g(Y) [X, Y] with some polynomial
g with integer coefficients, and (1) becomes
(3) nX"[X, Y]~g(Y) [X, Y] = 0, where f = n+t-ί>0 .
For any positive integers k, I, we denote by htι(X, Y) the polynomial (&+1) («*'
— ^ (Γ)") [JΓ, F]. By repeated use of (3), for any x, y<=R we have (£+!)«*'
j^ +*[*, j]=(Λ+l) n*'-1 «*[*, y] g(y)=nk'-1[xk+1, y] g(y}=nkl x< M^'[^+\ y\=(k+
1) «*' ^ +1)i/+*[ ,^ j]. Then, (Λ+l) «6' Λ;"+ί[Λ;, y]=(ft+l) «*'
1) WH Λ;"+*[Λ:, y] ^ "'=(^+1) «^+*^(j')w[*,3']. Therefore,
[x, y]=0, and hence ^/(ΛT, y)=0 (Lemma). In particular, w2[#, (1— ί
(I-*8") [*, j]=(«2-^(>')2) [*, J]=A21(^, y)-h12(x, y)=0, and therefore (l-^2")
[x, y]=Q Exchanging x and yy we have [Λ?, <y]=<y2// [xy y], which comes under
the case I). This completes the proof.
As an application of our theorem, we shall prove the following which in-
cludes [3, Theorem], [4, Theorem] and [5, Theorems 1 and 2].
Corollary 1. Let n>0, m, t and s be fixed non-negative integers such that
(n, t, m, s)Φ(l, 0, 1, 0). Suppose a left s-unital ring R satisfies the polynomial
identity
(4) X\Xn, Y]—[X, yn]i"s = 0 .
(a) If R has the property Q(n) then R is commutative.
(b) If n and m are relatively prime then R is commutative.
Proof. Let xy y be arbitrary elements of R, and choose an element e of
R such that ex=x and ey=y. If (m, s)Φ(l, 0) then (4) gives y=yen+eym+s
—ymeys<=yR. On the other hand, if (m, s)=(l, 0) then (n, t)*(\, 0) and (4)
gives x=xe~ xn+te+xn+t^xR. We have thus seen that R is s-unital. There-
fore, by [1, Proposition 1], we may assume that R has 1.
If m=0 (in the case of (a)), the assertion is clear by Theorem. Next,
COMMUTATIVITY THEOREM FOR RlNGS. II 813
we consider the case n=l. If m>0 and (m, *)Φ(1, 0) then m+s>l, and hence
the assertion is clear, again by Theorem. Also, if (m, s)— (1, 0) then, exchang-
ing the roles of X and Y, we get the assertion. Similarly, we can prove the
assertion for m=l. Therefore, we may assume henceforth that n>\ and
m>l. For the case (a), the assertion is immediate by Theorem. So, we
consider the case (b). Let a^D, and y^R. If a is not in C then there exists
a positive integer p such that ap&C and α*eC for all k>p and n[ap, y]=Q by
(2); similarly we can prove m[ap, y]=0. Hence, [ap, y]= 0. This contradic-
tion shows that DcΞC1, and (4) becomes
(5) nX*+'-l[X, Y]=mYm+s-ί[Xy Y] .
If n[x, y]=Q (x, y^R) then (5) gives mym+s~l[x, y]=nti*+*-l[x, y]=Q=nxn+t-1
[x, y-{-l]=m(y-}-l)m+s~~1[x, y], whence m[x, y]=Q follows by Lemma, and hence
[χy y] — O This prove that R has the property Q(n). Hence, R is commuta-
tive by Theorem, completing the proof.
Corollary 2. Let ri>Q and m be fixed non-negative integers. Suppose
a left s-unital ring R satisfies the polynomial identity [XY9 Xn+Ym]=Q. If
either R has the property Q(n) or n and m are relatively prime, then R is commu-
tative.
Proof. Actually, R satisfies the polynomial identity X[X", Y] — [X, Ym]Y
=0. Hence R is commutative by Corollary 1.
REMARK. In case n>0 and m=Q, Corollary 1 need not be true for right
^-unital rings (see [3, Remark]).
The author wishes to express his indebtedness and gratitude to Prof. H.
Tominaga for his helpful suggestions and valuable comments.
References
[1] Y. Hirano, Y. Kobayashi and H. Tominaga: Some polynomial identities and
commutativity of s-unital rings, Math. J. Okayama Univ. 24 (1982), 7-13.
[2] T.P. Kezlan: On identities which are equivalent with commutativity, Math. Japon.
29 (1984), 135-139.
[3] H. Komatsu: A commutativity theorem for rings. Math. J. Okayama Univ. 26
(1984), 109-111.
[4] E. Psomopoulos: A commutativity theorem for rings involving a subset of the ringy
Glasnik Mat. 18 (1983), 231-236.
[5] E. Psomopoulos: Commutativity theorems for rings and groups with constraints
on commutatorst to appear in Internat. J. Math. Math. Sci.
814 H. KOMATSU
Department of Mathematics
Osaka City University
Sumiyoshi-ku, Osaka 558
Japan


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