A NOTE ON CENTRALIZERS

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Internat. J. Math. & Math. Sci.

Vol. 24, No. 1 (2000) 55–57 S0161171200004245

A NOTE ON CENTRALIZERS

HOWARD E. BELL (Received3 January 2000)

Abstract.For prime ringsR, we characterize the setU∩CR([U,U]), whereUis a right ideal ofR; andwe apply our result to obtain a commutativity-or-ﬁniteness theorem. We include extensions to semiprime rings.

Keywords and phrases. Prime rings, semiprime rings, centralizers.

2000 Mathematics Subject Classiﬁcation. Primary 16N60, 16U80.

LetR be an arbitrary ring with centerZ. Forx,y∈R, denote by[x,y]the commutator xy−yx; andfor an arbitrary nonempty subset S of R, denote by [S,S]

the set{[x,y]|x,y∈S}. Denote byCR(S) the centralizer ofS inR—i.e., the set {x∈R|[x,s]=0 for alls∈S}.

It is provedin [2] that ifRis semiprime andIis a nonzero ideal ofR, thenCR([I,I])⊆ CR(I). It follows thatC([I,I])∩I ⊆Z, since in a semiprime ring R the center of a nonzero right ideal is contained in the center ofR. The ﬁrst goal of this note is to study the subringH=CR([U,U])∩U, whereRis prime or semiprime andU is a nonzero right ideal. The information obtainedis usedto prove commutativity-or-ﬁniteness results extending [1, Theorem 3].

1. Preliminaries. We shall use standard notation for annihilators—that is, for a nonempty subsetS ofR,Al(S)andA(S)will be the left andtwo-sidedannihilators ofS. A subringS will be saidto have ﬁnite index inR if(S,+)is of ﬁnite index in (R,+). We shall use without explicit mention the commutator identities [xy,z]= x[y,z]+[x,z]yand[x,yz]=y[x,z]+[x,y]z.

We begin with a revealing example.

Example1.1. LetF be an arbitrary ﬁeld, letR be the ring of 2×2 matrices over F, andletU=e11R. ThenRis prime,U is a right ideal, and[U,U]=Fe12. Note that CR([U,U])∩U=Fe12=A([U,U])∩U, andnote that this set does not centralizeU.

Thus, the result in [2] for two-sided ideals does not hold for one-sided ideals, even in the case of prime rings.

2. The case ofRprime

Theorem2.1. LetRbe a prime ring,Ua right ideal ofR, andH=CR([U,U])∩U.

Then eitherH=U∩Z, orHis a zero ring andH=A([U,U])∩U. In any case,His a commutative subring ofR.

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56 HOWARD E. BELL

Proof. We begin as in the proof of [2, Lemma 1]. Letz∈CR([U,U]). Then for all x,y∈U,z[x,xy]=[x,xy]z; hencezx[x,y]=x[x,y]z=xz[x,y]andtherefore [z,x][x,y]=0. Replacingy byyz, we get [z,x]U[z,x]= {0}for all x∈U; and since[z,x]Uis a nilpotent right ideal, we have[z,x]U= {0}for allz∈CR([U,U]) andx∈U. Takingz∈H, we obtain[z,x]z=0=z[z,x]for allz∈Handx∈U; and replacingxbyxrfor arbitraryr∈RyieldszU[z,r ]= {0}, hence

zUR[z,r ]= {0} for allz∈Handr∈R. (2.1) SinceR is prime, (2.1) shows that eitherz∈Z orzU = {0}; henceH=(H∩Z)∪ (H∩Al(U)). Since the abelian groupHcannot be the union of two proper subgroups, we haveH=H∩ZorH=H∩Al(U), so thatH⊆ZorH⊆Al(U). In the ﬁrst case, His clearly equal toU∩Z, so supposeH⊆Al(U). SinceH⊆U,H²= {0}; moreover, H⊆Al([U,U])∩CR([U,U]), soH⊆A([U,U])andhenceH=A([U,U])∩U.

We now proceedto a commutativity-or-ﬁniteness result.

Theorem2.2. LetRbe a prime ring andUa right ideal of ﬁnite index inR. If[U,U]

is ﬁnite, thenRis either ﬁnite or commutative.

Proof. Suppose that[U,U]= {x1,x2,...,xm}. For eachi=1,2,...,mdefineΦi: U→UbyΦi(x)=[xi,x]for allx∈U. ThenΦi(U)is finite, hence KerΦiis of finite index inU. LettingH=_m

i=1KerΦi, we see thatH=U∩CR([U,U])andthatH is of finite index inU. NowUis of finite index inR, soHis of finite index inR. It follows by a theorem of Lewin [3] thatHcontains an idealIofRwhich is also of finite index inR. IfI= {0}, thenRis finite; ifI≠{0}, Theorem 2.1 implies thatRhas a nonzero commutative ideal and henceRis commutative.

3. The case ofR semiprime. LetR be semiprime, U a right ideal, and H=U∩ CR([U,U]). Let{Pα|α∈Λ}be a collection of prime ideals such that∩Pα= {0}. Now (2.1) holds inR, hence for eachα∈Λandeachz∈H, either[z,R]⊆PαorzU⊆Pα. Since each of these conditions deﬁnes an additive subgroup ofH, we see that[H,R]⊆ PαorHU⊆Pα; therefore[H,H]⊆Pαfor allα∈Λ. Thus[H,H]= {0}—that is,His a commutative subring ofR.

Revisiting the proof of Theorem 2.2, we see that in the semiprime case, eitherR is ﬁnite or R contains a nonzero commutative ideal I. But in a semiprime ring, a commutative ideal is central; hence we have the following extension of Theorem 2.2.

Theorem3.1. LetRbe a semiprime ring andUa right ideal of finite index inR. If [U,U]is finite, then eitherRis finite orRcontains a nonzero central ideal.

References

[1] H. E. Bell andA. A. Klein,On rings with Engel cycles. II, Results Math.21(1992), no. 3-4, 264–273. MR 93d:16039. Zbl 786.16015.

[2] M. N. Daif andH. E. Bell,Remarks on derivations on semiprime rings, Internat. J. Math.

Math. Sci.15(1992), no. 1, 205–206. CMP 1 143 947. Zbl 746.16029.

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A NOTE ON CENTRALIZERS 57 [3] J. Lewin,Subrings of ﬁnite index in ﬁnitely generated rings, J. Algebra5(1967), 84–88.

MR 34#196. Zbl 143.05303.

Howard E. Bell: Department of Mathematics, Brock University, ST. Catharines, On- tarioL2S 3A1, Canada

E-mail address:[email protected]