Also weprove that if R is a2-torsion free ring with in which (2k)n+1-(z2k)n5Z(R)for allzeR and fixed positiveintegeraandnon-negative integer k, then R is commutative

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 2 (1997) 409-411

409

TWO

ELEMENTARY COMMUTATIVITY THEOREMS

FOR

GENERALIZED

BOOLEAN RINGS

VISHNU GUPTA

Departmentof Mathematics M.D. University,P.G. RegionalCentre

Rewari

(Haryana),

India

(Received

September 9, 1991 andin revisedform April 17,

1992)

ABSTRACT.

In

this paper weprove thatif R is aring with as an identity elementin which

r

m- xr’e

Z(R) for all xeR and fixed relatively prime positive integers ^m and n, oneofwhich is even, then R is commutative. Also weprove that if R is a2-torsion free ring with in which

(2k)

ⁿ⁺

1-(z2k)

ⁿ5Z(R)for allzeR and fixed positiveintegeraandnon-negative integer k, then R is commutative.

KEY

WORDS AND

PHRASES. Commutator,

^2-torsionfree ring.

1991AMS SUBJECT

CLASSIFICATION

CODE. 16A70.

1.

INTRODUCTION.

Throughout this paper, R is anassociative ringwith as anidentity element. We denote the centre ofR by Z(R) and the commutatorzy-yz by [z,y]. Recently, Quadri and ^As]arM

[1]

provedthat if Ris aringin which x

+

^1_

zn

eZ(R)for allxeR and fixed positiveintegern,then R is commutative. Inthis paper, wegeneralizethisresult.

2. MAIN RESULTS.

Westartwiththefollowinglemma of Bell

[2].

LEMMA

2.1. Let tveR. If for each xeR there exist relatively prime positive integers

n n(z)andm re(z)^{such that}

[w,z

n]

[w,z

m]

0, thenteZ(R).

THEOREM

2.1. IfR is a ringwith z

rn- zne

z(R) for M1 ^zeR dfixedrelatively prime positiveintegersmdn, oneof which is even,thenR iscommutative.

PROOF. Let

zm-_zⁿeZ(R) for 1zeR.

(2.1)

Asse miseven dnisodd. Using bothz d -zin

(2.1)

^{d then} adding d subtracting, weget

2zm

Z(R) d

2zn

Z(R). Thus [zm,2y]=[zn,2y]=O for r,y R; d byLena 2.1 2y Z(R)for y qR. Nowwereplacerby

+

^toobtMn

d since ^miseven dnis d d [,]

,

^we

ge [()-

,1 forsome()Z[]. Now

410 V. GUPTA the theoremfollows fromHerstein’sresult

[3].

In

Theorem 2.1, allthehypotheses areessential. If both ^mand nareodd orifoneofrnand

nisevenand the otherodd,but theyarenot relativelyprime; orifbothmandnareeven; orif R is a ring without the identity element in the hypotheses of thetheorem, then / need not be commutative.

EXAMPLE

2.1. Itcanbe shown easily that

R 0 a d :a,b,c,d 6.GF(3

0 0 a

isaringwithidentityelement,in which

(i)

z3_z9 e

(ii)

^z

3-z

⁶e

(iii)

^{z4 z1}6-g(R)

forallz 6-R, butR isnot commutative.

EXAMPLE

2.2.

R=

0

a d :a,b,c,d 6.GF(2)

0 a

isaringwithidentity elementin which

(i)

^{z z9}6-Z(R)

(ii)

^z4

zS

6-

for allr6-R, butRisnot commutative.

EXAMPLE

2.3.

l/

R= 0

0

0 :a,b,c,6-GF(3)

0 0

isaringwithoutidentity elementwithz3 _z4

6-Z(R) for allz6-R, butR is not commutative.

Westate thefollowinglemmawhichcanbe proved easily.

LEMMA

2.2. If

2kn

wherekandnarepositive integers,

hen

aremultiples of2

:

(2r 1)

+

^2k r k2r 1/

for 1,2,3,

-.

Now

wegive thefollowing ^theoremwhich generalizes thetheoremof Quadri and Ashraf

[1]

for2-torsionfree rings.

THEOREM 2.2. IfR is a 2-torsionfree ring with

(z2k)

ⁿ

⁺ _(z2k)n

6-Z(R) for all r6-R and fixednon-negative integerkandpositiveintegern,thenR iscommutative.

PROOF.

Ifk 0thenresultfollows from Theorem2.1. Letk>0and [z

+ 2k,

p]

[zt,

u]^{for all}z,u Rwhere

2/%

Now

wereplacezbyz

+

^{to obtain}

COMMUTATIVlTY THEOREMS FOR GENERALIZED BOOLEAN RINGS 411

(2.2)

Nextwereplace ^zby-zin

(2.2)

^andsubtract theresult from

(2.2)

andusethefact that R is 2-torsion free toget

Lr

^k2r-II

+r =ll

^{\(2r-1)+2k r-1}

By

Lemma 2.2 and thefact that R is 2-torsionfree, ^weget

[z2p(x)-x,y]

^{0 for all z,y}ER and somep(x)EZ[x]. NowRiscommutative.

All thehypotheses ofTheorem 2.2 areessential.

In

Example2.1, R is a2-torsion free ring withidentityelement in which

(x2k)m- (x2k)

ⁿ Z(R) (k 2,m 4,n 7)for allx,yER andmandn are relatively prime positive integers andone of them is even, but R is not commutative.

In

Example 2.2, R isa 2-torsionringwith identity elementin which

(x2/)

^n/1-

(x2/c)n

Z(R)(k-2,

,

_{1) for allx R, but} R is not commutative.

In

Example 2.3, R is a2-torsionfree ringwithout identity element in which

(x2t)n+l-(x2k)nEZ(R)

(k=2 and n=l) for all zR, but R is not commutative.

ACKNOWLEDGEMENT. The author expresses his sincere thanks to the referee for many helpful suggestionstomodifytheproof of Theorem2.1.

REFERENCES

1.

QUADRI,

^M.A.

& ASHRAF, MOHD.,

Commutativity ofgeneralized Boolean rings, Publ.Math.

(Debrecen)35 (1988),

73-75.

2.

BELL, H.E.,

Onringswithcommutingpowers, Math. Japonica 24

(1979),

473-478.

3.

HERSTEIN, I.N.,

Tworemarksonthe commutativityof rings, Cand.J. Math. 7

(1955),

411-412.

4.

HERSTEIN, I.N., A

generalization of theorem ofJacobson, Amer. J.^{Math. 73}

(1951),

^755-

762.