Mathematical Problems in Engineering

IJMMS 2004:54, 2863–2865 PII. S0161171204405420 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

A COMMUTATIVITY-OR-FINITENESS CONDITION FOR RINGS

ABRAHAM A. KLEIN and HOWARD E. BELL Received 20 May 2004

We show that a ring with only finitely many noncentral subrings must be either commutative or finite.

2000 Mathematics Subject Classification: 16U80, 16P99.

1. Preliminaries. In [1] the following theorem was proved.

Theorem1.1. IfRis a periodic ring having only finitely many noncentral subrings of zero divisors, thenRis either finite or commutative.

In view of this result, it was conjectured that any ring with only finitely many non- central subrings is either finite or commutative. It is our principal goal to prove this conjecture; and in the process we provide a new proof ofTheorem 1.1.

For any ringR, the symbolsN,D,Z, and℘(R)will denote, respectively, the set of nilpotent elements, the set of zero divisors, the center, and the prime radical; andR will be called reduced ifN= {0}.

We will require the following three lemmas.

Lemma1.2[5, Corollary 5]. IfRis any ring in whichNis finite, thenR/℘(R)=B⊕C, whereBis reduced andC is a direct sum of finitely many total matrix rings over finite fields.

Lemma1.3. IfRis any ring in which℘(R)⊆Z, then zero divisors inR/℘(R)may be lifted to zero divisors inR.

Proof. We show that if ¯x=x+℘(R)is a zero divisor in ¯R=R/℘(R), thenx is a zero divisor inR. Clearly, this is the case for ¯x=¯0, so we assume that ¯x≠¯0 and that ¯y∈R¯\ {¯0}is such that ¯xy¯=¯0. Then,xy∈℘(R), and there exists nsuch that (xy)ⁿ=0. Moreover, sincexyr∈℘(R)for allr∈R, we havexy∈Zandxyr∈Z for allr∈R.

Sincey∈℘(R),yis not strongly nilpotent, therefore there existr1,r2,...,rn−1∈R such thatyr1yr2···yrn−1y≠0. Sincexyand allxyriare central, we getxⁿyr1yr2

···yrn−1y=xyr1xⁿ⁻¹yr2···y=xyr1xyr2xⁿ⁻²···y=xyr1xyr2···xyrn−1xy

=(xy)ⁿr1r2···rn−1=0; hencexⁿis a zero divisor inR, and so isx. Lemma1.4[3]. IfRis a periodic ring withN⊆Z, thenRis commutative.

2. The main theorems. We begin with results on rings having only finitely many noncentral subrings of zero divisors, which we will callD-nearly central rings (DNC- rings).

2864 A. A. KLEIN AND H. E. BELL Lemma2.1. IfRis any DNC-ring, thenN⊆ZorNis finite.

Proof. AssumeN\Z≠φand leta∈N\Z. By [1, Lemma 3],ahas finite additive order and henceagenerates a finite noncentral subring of zero divisors. Thus,N\Zis finite; and sinceu+N∩Z⊆N\Zfor allu∈N\Z, we see thatN∩Zis also finite.

Lemma2.2. LetRbe an infinite DNC-ring withNfinite. Then every nil ideal ofR is central.

Proof. SupposeIis a noncentral nil ideal. SinceIis finite, the two-sided annihilator A(I)is of finite index inR, hence is infinite. Moreover, for any subringSofA(I),S+Iis a noncentral subring of zero divisors ofR; thus, there are only finitely many subrings S+IwithS a subring ofA(I). It follows at once thatA(I)has only finitely many finite subrings, since otherwise there exist infinitely many distinct finite subringsS1,S2,..., such thatS1+I=S2+I=··· =Sn+I= ···is a finite ring with infinitely many subrings.

It is well known that every infinite ring has infinitely many subrings (cf. [6]); hence A(I)has infinitely many infinite subrings, and so does every infinite subring ofA(I). ThusA(I)has an infinite strictly decreasing sequenceS1⊃S2⊃ ··· of subrings; and for somej,Sj+I=Si+Ifor alli≥j. For eachi≥j, chooseai∈Si\Si+1and write ai =bi+1+ci, where bi+1 ∈Si+1 and ci ∈I. Since I is finite, there exist h, k such thatj≤h < k andch=ck. It follows thatah=bh+1+ak−bk+1so that ah∈Sh+1, a contradiction. Therefore,Imust be central.

Lemma2.3. IfRis any DNC-ring, then eitherRis finite orN⊆Z.

Proof. LetRbe any infinite DNC-ring. By Lemmas2.1and2.2, we may assume that Nis finite and℘(R)⊆Z; and it follows byLemma 1.3that ¯R=R/℘(R)is a DNC-ring.

ByLemma 1.2, ¯R=B⊕C, whereBis an infinite reduced ring andC is a direct sum of finitely many total matrix rings over finite fields. SupposeC ≠{0}and contains the total matrix ringKnover the finite fieldK. We may assumen >1; otherwiseKncan be incorporated intoB. NowBhas infinitely many subrings; and for each subringS ofB, S+Ke1nis a noncentral subring of zero divisors of ¯R. Since ¯Ris a DNC-ring, we must therefore haveC= {0}so that ¯Ris reduced. It follows that inR,N=℘(R)⊆Z.

Remark2.4. Theorem 1.1follows at once from Lemmas1.4and2.3. Thus, we have produced a proof ofTheorem 1.1which is very different from the one in [1].

We now arrive at our main result.

Theorem2.5. IfRis any ring with only finitely many noncentral subrings, thenRis finite or commutative.

Proof. SupposeRis a counterexample with the minimum possible number of non- central subrings. ThenRis infinite and every proper infinite subring ofRis commuta- tive. SinceRis a DNC-ring, Lemmas1.4and2.3show that all finite subrings ofRare commutative as well; henceRis a so-called one-step noncommutative ring—a noncom- mutative ring in which every proper subring is commutative.

Note that ifHis any set of pairwise noncommuting elements ofR, the centralizers of the elements ofHare pairwise distinct noncentral subrings ofR. Hence any set of

A COMMUTATIVITY-OR-FINITENESS CONDITION FOR RINGS 2865 pairwise noncommuting elements is finite, and by [2, Theorem 2.1],Zhas finite index inR. But by a theorem of Ikeda [4], a one-step noncommutative ring with[R:Z] <∞ must be finite. Thus, we have contradicted our assumption thatRwas a counterexample.

Acknowledgment. This work was supported by the Natural Sciences and Engi- neering Research Council of Canada, Grant 3961.

References

[1] H. E. Bell and F. Guerriero,Some conditions for finiteness and commutativity of rings, Int. J.

Math. Math. Sci.13(1990), no. 3, 535–544.

[2] H. E. Bell, A. A. Klein, and L. C. Kappe,An analogue for rings of a group problem of P. Erd˝os and B. H. Neumann, Acta Math. Hungar.77(1997), no. 1-2, 57–67.

[3] I. N. Herstein,A note on rings with central nilpotent elements, Proc. Amer. Math. Soc.5(1954), 620.

[4] M. Ikeda,Über die einstufig nichtkommutativen Ringe, Nagoya Math. J.27(1966), 371–379.

[5] A. A. Klein and H. E. Bell,Rings with finitely many nilpotent elements, Comm. Algebra22 (1994), no. 1, 349–354.

[6] T. Szele,On a finiteness criterion for modules, Publ. Math. Debrecen3(1954), 253–256.

Abraham A. Klein: School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

E-mail address:[email protected]

Howard E. Bell: Department of Mathematics, Brock University, St. Catharines, ON, Canada L2S 3A1

E-mail address:[email protected]

Mathematical Problems in Engineering

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Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

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Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil;

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Elbert E. Neher Macau,

Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected]

Center for Applied Dynamics Research, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected]

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