Bull. Kyushu Inst. Tech.
(M. & N, S,) No. 19, 1972
A REMARK ON THE LIE STRUCTURE OF
ASSOCIATIVE RINGS By
Eishi HoNGo and Mituru ODA
(Received October 15, 1971)
1. Introduction
If R is any associative ring we can introduce in it the structure of the Lie ring with the addition as defined in R and the new multiplication [x, or]=xy
- orx for all x, or in R, where xyis the original associative product of R. An ad- ditive subgroup Uis said to be a Lie ideal of R if for any elements u in U and x in R the commutator [u, x] is again an element of U. If X and Y are two subsets of R we denote by [X, Y] the additive subgroup generated by the ele- ments of the from [x, or], where x and or range over the elements of X and y respectively.
An associative ring R is said to be semiprime if its only nilpotent ideal is the zero ideal, and R is said to be 2-torsion-free if in it 2x =O implies x==O. In all that follows we always denote by Z the center of R and by (O) the zero ideal or the set consisting only of the zero element.
In his investigation of the Lie structure of an assoeiative ring, I. N. Her- stein [1] proved the following two lemmas:
(I) Let R be a semiprime, 2-toTsion-free ring ancl let U be a Lie ieleal of R.
Suppose that [U, U](Z, then U(Z (Lemma 1).
(II) Let R be a semiprime, 2-toTsion-free ring ana let U be a Lie ideal of R.
Szepupose that A( U is an aaditive szebgToup such that [U, A](A and [A, A](Z.
Then [A, U]=(O) (Lemma 4).
The purpose of this paper is a further study of the above lemmas (I) and (II). In the second section we prove a theorem which is, in a sence, a generali- zation of (I) and (II) by a slight modification of the proof of (II). In the third
section we offer the lemmas (I) and (II) as corollaries of our theorem with the aid of some results of Herstein [1].
2. Theorem
THEoREM. Let R be a semipTime, 2-torsion-free associative ring and let U
16 E. HoNGo and M. ODA
be a Lie nteal of R. Then for an element t of R, [t, [t, U]](Z if and only if
[t, [t, u]] =- (o).
PRooF. The "if" part is trivial, so it is sufficient to prove the "only if"
part. Lettbe such an element of R that [t, [t, u]](z.
We suppose
[t, [t, u]]l(o),
then there exists an element u of U such that [t, [t, u]]EZ, [t, [t, u]]=7EO.
Now we put
[t, u] =a, [t, [t, u]]==[t, a]=b, [[t) u]) t2[t) u]] == [a) t2a]=v
for brevity. Since Uis a Lie ideal of R, we have
ac U, bE UA Z, vE U.
In our calculations of commutators we always use these three properties and the identity
[x, y] == [x, or]z+ or[x, x].
First, we note that
av=a[a, t2a] == [a, at2a] c U Then from the assumption [t, [t, U]](Z we obtain Z) [t, [t, av]]=:[t, [t, a]v+a[t, v]]
== [t, 6]v+2b[t, v]+a[t, [t, v]]
=2b[t, v]+a[t, [t, v]].
Therefore, commuting this with a, we have O=[a, 2b[t, v]]+[a, a[t, [t, v]]]
=2[a, b][t, v]+2b[a, [t, v]]+[a, a][t, [t, v]]+a[a, [t, [t, v]]]
==2b[a, [t, v]].
ARemark on the Lie Structure of Associative Rings 17
Since R is 2-torsion-free, the above gives
b[a[t, v]] =O. (1)
We next note that [t) v]=[t) [a) t2a]]
=[t, [a, t]ta]+[t, t[a, t]a]+[t, t2[a, a]]
== [t, -bta]+[t, -tba] == -2[t, tba]
== -2([t, t]ba+t[t, 6]a+tb[t, a]) == -2tb2.
Therefore, commuting [t, v] with a, we have [a, [t, v]]=[a, -2tb2] =-2([a, t]b2+t[a, b2]) = -2b2[a, t] == 2b3.
So we get
b[a[t, v]]=2b`. (2)
It follows from (1) and (2) that 2b4=O. Since R is 2-torsion-free, this im- plies
b4==O.
JNow we consider the ideal bR of R. The above gives
(bR)4=b4R=:(O).
Therefore bR is a nilpotent ideal of R. Since R is semiprime we conclude b==O.
This contradicts the assumption b=[t, [t, u]]4 O. Thus the proof is completed.
3. Corollaries
CoRoLLARy 1. Let R be a semipTime, 2-toTsion-free assoeiative Ting anel let U be a Lie uleal of R. Smppaose that [U, U] ( Z, then U( Z.
PRooF For uE U, xcR we have
[u, [u, v]] E [Ul, U] ( Z.
Therefore, from our theorem we get
18 E, HoNGo and M. ODA
[u, [u, x]] ==O.
It follows from this and Sublemma in Herstein [2, p. 5] that u E Z.
CoRoLLARy 2. Let R be a semiprime, 2-toTsion-free associative ring and let U beaLie ideal of R. Szeppose that A is an additive s2ebgToup such that [U, A](A and [A, A](Z. Then [A, U]==(O).
PRooF. For tEA wehave
[t, [t, U]]([A, A](Z.
Therefore, from our theorem we get [t, [t, u]]==(o).
It follows from this and Theorem 1 in Herstein [1] that [t, u]-(o).
We finally note that Corollary 1 is quite the same as (I) but Corollary 2 is a little different from (II). In our proof of Corollary 2 there was no need to as
sume A( U, so we removed the assumption A(U from (II). From this point of view Corollary 2 is a generalization of Lemma 4 in Herstein [1].
Addendum
Professor K. Yamaguti (Kumamoto University) pointed out to us that the theorem holds for [t, [t, •••, [t, U]]•••] under some additional assumption. In the forthcoming paper we will discuss this material.
References
[1] I.N. Herstein, On the Lie structure of an associative ring, Journal of Algebra, vol. 14 (1970), 561-571, [2] I.N. Herstein, "ToPics in Ring Theorpt", University of Chicago Press, Chicago, 1969.
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