Generalized periodic rings

Internat. J. Math. & Math. Sci. VOL. 19 NO. (1996) 87-92

87

GENERALIZED PERIODIC RINGS

HOWARD E.BELL

Department

of Mathematics

BrockUniversity St.Catharines

Ontario,Canada

and

ADILYAQUB

Department

ofMathematics

University ofCalifornia

SantaBarbaraCA, U.S.A.

(Received March 18, 1994 and in revised form March 13, 1995)

ABSTRACT. Let R be aring, andlet N and C denotethe setof nilpotents and thecenterof R, respectively. Riscalled generalized periodiciffor every x eR\

(N

u

C),

there exist distinctpositive integers m, nof opposite parity suchthat

x"

x"

eN cC. We provethatageneralized periodic ring alwayshas theset N of nilpotents forming anidealin R. Wealsoconsidersome conditionswhich

imply the commutativity of a generalized periodic ring.

KEY WORDS. AND PHRASES Commutativity, periodic ring, generalized periodic ring,centerof a

ring,commutatorideal.

1991AMS SUBJECTCLASSIFICATION CODES. 16U80,16D70.

1. INTRODUCTION.

Throughoutthepaper, Rwilldenotearing,Nthesetof nilpotents,Cthecenter, J the Jacobson radical, and

C(R)

thecommutator idealofR. Thering

R

iscalled periodicifforevery x in

R

there

exist distinctpositive integersm, n such that x x*.

An

elementxofRiscalled potent if, for some positive integern> 1,

x"

x.

R

iscalled weakly periodicifeveryelement x of

R

canbewrittenasa

sumof a potent element andanilpotent element. Itiswell known thataperiodic ringisnecessarily weakly periodic. Whetheraweaklyperiodic ringisnecessarily periodicisapparentlynotknown,except inthepresenceof other additionalhypotheses. Wenowformallystatethedefinition of ageneralized periodicring.

Definition.

A

ringRiscalledgeneralizedperiodicifforeveryx in

R,

x N u C, wehave

x*

x"

N n

C,

for somepositiveintegers m, nof opposite parity.

Or,

equivalently,

x*-x*+k

NnC;

n,kZ/;

kodd;(xNC).

(1.1)

We provethat thesetof nilpotentsinageneralizedperiodic ring Risalways anideal inR.

We

alsoconsider conditions whichimplythe commutativityofa generalizedperiodic ring.

2. STRUCTUREOF GENERALIZED PERIODIC RINGS. Webegin with some basic facts about generalized periodic rings.

Lemma

1.

In

a generalized periodic ringR,wehave

(i)

C(R)

J; (ii)

J

c2N)C; (iii) N J.

PROOF (i).

By

awell known theoremofHerstein[1],if

R

isa division ring which satisfies(1.1), then

R

is commutative.

Next,

supposethat

R

isaprimitive ringwhich satisfies(1.1). Since(1.1)is inheritedby all subrings ofRandby all homomorphic images ofR,itfollows, byJacobson’sDensity Theorem, thatif

R

is notadivisionring, then some completematrixring

D,,,

with m>1, over a division ring

D

satisfies(1.1). This,however, is false, as can be seenbytaking x

El2

+

E21

in

Dm.

Hence,

a primitive ring which satisfies(1.1)isnecessarilya division ring, and henceiscommutativeby Herstein’sTheorem. Therefore,any semisimplering which satisfies(1.1)is commutative, whichproves 0).

(ii).

Suppose

x

J,

x

N,

x C. Then, by (1.1), x x

N,

n m, and thus for some q

z

/.

g(x)

xq

x’+’g(x);

(g(g)

Z[X]).

_(2.1)

It is ,eadily ,,erified that

e=[g(,0]

is an idempotent element in J

(since

x

J),

and hence

set, by and hence contradiction. This contraction

proves (ii).

(iii). First, weprovethat

axN forallaeN and all xR. (2.2)

Toshow this,first notethatby(i) and(ii),

C(R)NuC.

(2.3)

Suppose

(2.2) isfalse, and let a

N,

x

R,

ax N

(for

some a and

x).

Let

R

/

C(R),

and let

x

+

C(R)

beanarbitrary element of

.

Since iscommutative,(2.2)implies that

’

isnilpotent’ and hence

(ax)’

C(R)

forsome positive integer r. Thus,by(2.3)

(ax)

Nor

(ax)

C. Since,by hypothesis, ax

N,

therefore

(ax)

Cfor somepositiveinteger r.

Since a

N,

leta 0. Notethat,since

(ax)’

C,

(ax)

(axy

a.

(axy-(xa)

-lx

a2xtl

(some

t

R).

Continuingthisprocess,weseethat

[(axyp

akktk_x

(some

tk_t

R),

forall k

Z

/.

In

particular,

[c r]

a xte_ 0

( in a

0),

and hence ax

N,

contradiction. This contradictionproves (2.2).

To

completetheproofof(iii), let a

N,

x

R.

Then, by(2.2),ax N and hence ax is rightquasi-regularfor all x in

R,

whichimplies a

J.,

Thiscompletestheproofofthe lemma.

We

are nowin apositiontoprovethefollowing fundamentaltheorem.

GENERALIZED PERIODIC RINGS 89

PROOF.

By

Lemma (iii), (ii), we have

N J N u C. (2.4)

Let aN,beN. Then aeJ,beJ, a-beJ, and hence [see (2.4)] a-beN or a-beC. If

a-beC, then ab=ba and hence a-beN. So, in any case, a-beN for all aeN, beN. We

have already established [see 2.2)] that axeN forall a eN,xeR, and a similar argument yields

xa eN. Therefore, Nis anideal.

THEOREM 2. Let R be ageneralized periodic ring. Then R/N is commutative, and hence

C(R)

; N.

PROOF

By

Theorem 1,Nisan ideal, andhenceR/Nmakessense. Let x eR,x C. Then, by (1.1),

x"

x eN, for some n>m, say. (2.5) Itisreadilyverifiedthat

(x

"-’+

x)

(x

*-"*/

x)xm-g(x),

some

g()e

Z[L],

and hence

x"-m+’ x eN

(since

x"

x e

N).

Therefore,for all x eR,wehave

x-x"-m+eN or xeC, n>m,

(xR).

(2.6) Hence, R/Nhastheproperty thatforeach x

R

/N,thereexists k> forwhich x

xk

iscentral.

By

a wellknown theoremof Herstein ],itfollows thatR/Nis commutative,and thus

C(R)

t:::N.

Since N isanidealof R(Theorem 1), therefore NGJ. Combiningthis with

C(R)

N and

Lemma (ii) weobtain

LEMMA

2. Let Rbeageneralized periodic ring. Then

C(R)

N JtN u C. (2.7)

Next,weconsider aringwhich isboth weakly periodic and generalizedperiodic.

THEOREM3. Ifaring Risboth generalized periodicandweakly periodic, then Risperiodic. PROOF. Let x eR. Since Risweaklyperiodic,wehave

x=a+b for some aeN, bpotent

(bq

=b,

q>l).

(2.8) Thus, x-a

(x-a)q;

and since N is an ideal, we have

x-xq

N.

By

a well known theorem of Chaeron[2],itfollows thatRisperiodic.

3. COMMUTATIV1TYOF GENERALIZEDPERIODIC RINGS.

Wenowturnour attentiontosomeconditionswhich, whenimposedon ageneralized periodic ring, renderit commutative. Webeginwiththefollowing result,which issuggested byanoldtheorem

onperiodic rings.

THEOREM 4.

Let R

be a generalized periodic ring, and suppose N C. Then

R

is

commutative.

PROOF.

By

(2.6),for each x

R,

either x eCor x-x eN for some k>1. Since N C, therefore, forevery x

R,

x x C for some k>1. Therefore, byHerstein’sTheorem 1],

R

is

commutative.

COLLARY 1.

Let R

be a generalized periodic ring, and suppose J C. Then

R

is commutative.

PROOF. By

Lemma2,N

J,

andhence N C. Thus,

R

iscommutative,byTheorem 4. COLLARY2. Let

R

beageneralized periodic ringwithJacobsonradical J. ThenJ Nor R

is commutative.

PROOF.

By

Lemma (ii),itfollows that

Viewing (3.1)as a relationholdingonadditivesubgroups,we conclude that

J=JN or J=JC. (3.2)

Thisimplies that

J

N

or Jc:C. (3.3)

IfJ N, then J N [see (2.7)]. Ontheotherhand,ifJc;C, then R iscommutative,byCollary1.

COROLLARY3. Let R be ageneralizedperiodic ring which is notcommutative. Then J

coincides with N.

Beforestatingthenexttheorem,let us f’n’st considerthe followingtwoexamples,whichshowthat neithercentralityofidempotentsnorcommutativityofnilpotentelementsimplies commutativityof a

generalized periodic ring. Notethat,ineachcase,centralelementsare zero divisors.

EXAMPLE1. Let

R={(

),

(:

)/

:3’

(

ll)]0’IGF(2’}"

ItisreadilyverifiedthatRis ageneralizedperiodic ringwithcommuting nilpotents butitsidempotents

arenotinthecenter.

EXAMPLE

2. Let

R= 0

a,

b, c,

GF(3)

0

Itcanbeseenthat Ris,againageneralized periodic ringwithcentral idempotents butitsnilpotentsdo not commutewitheach other.

Experience shows thata condition whichdoesnotimply commutativity forgeneralringsmaydo

sofor ringswith1. Indeed,wecanshow that generalized periodic ringswith are commutative; infact,

inthefollowing theorem,wecando better than that.

THEOREM5. SupposethatRisageneralizedperiodic ring containingacentralelement which isnota zero divisor. Then Ris commutative.

PROOF. In viewof Theorem4, we needonly show that N C. Suppose not,and choose

a0 N\C. Let (0> be theminimal positive integer for which

a

C for all >(0; and let

a=a

-.

Note that aC, and a

xC

for all >2. Now if cC is not a zero divisor, then

c

+

a N C,so thereexist n, mof opposite paritywith n>m,such that

(c

+

a)*

(e

+

a)"

nc,

(n

>

m).

(3.4)

Weshall assume thatnisevenand m isodd, the other case being only marginallydifferent.

From (3.4) wehave

nc"qa-mc-a

C, fromwhich it follows that(since c is not azero

divisor)

nc’-%-

ma C. (3.5)

Anotherconsequenceof(3.4) isthat

c"-c

N and hence c -c N, where istheeveninteger n-m+l. Replacing c by-c in our argument, we also get an even integer k such that

(-c)-(-c)N.

Since N is an ideal, we have

c/’0-)-cN

and

(-c)/’(-)-(-c)eN

for all positive integers s and t; and taking

q=l+(j-1)(k-1),

we see that q is even,

ca-cN

and

(-c)-(-c)

N. Itfollowsatoncethat 2cN and hence 2c=0 for some positive integerr. Sincecisnota zerodivisor,thisyields 2R

{0};

and,inparticular,

2 a C. (3.6)

GENERALIZED PERIODIC RINGS 91

ma

2n0cn-ma

+z1, z EC. (3.7)

Therefore, using (3.7)wesee that

m2a

2n0cn-mma

+mz

2n0cn-m

(2n0cn-ma

+z

)+

_mzl

=22(n0cn-m)2a+z2,

z eC;

andproceedinginductively,we get(see(3.6))

mra

C. (3.8)

Since m wasodd,(3.6)and(3.8)areincompatiblewiththe assumptionthat ae C. Therefore N

_

C, as

required. Thisprovesthetheorem.

COLLARY4. Let Rbe a ring with 1. IfRisgeneralizedperiodic, thenRiscommutative.

COLLARY5. Let Rbe aprime ringwith nonzerocenter. IfRisgeneralizedperiodic,thenRis commutative.

Ourfinaltheoremconfrontsthe impediments ofExamples and2in a more directway.

THEOREM6.

Suppos

Risageneralized periodic ring, Nthesetofnilpotents, andE thesetof idempotents of R. Supposethat

(i) E C

(center

of

R);

and

(ii)

Every

commutator

[a, b]

ab- ba with aeN and b N is potent

(i.e.,

[a, b]

q=

[a, b]

for some q >

1).

Then Ris commutative.

PROOF.

By

(2.7),

C(R)_

N, andhence

[a, b]

N. By hyothesis,

[a, b] [a, b] =[a,b]

/x(q-) for all positive integers

,,

and hence

[a,

b]

0

(sin

ce

[a, b]

E

N).

Thus,

[a,b]=0

for all a, b eN i.e., N iscommutative. (3.9)

Recall alsothat,in(2.6),weproved that,for ever x in R,wehave

x-xkeN

for some k>l, or xeC,

(xR).

(3.10)

Combining(3.9),(3.10),wesee that

For all x,y in R,

Ix

x

k,

y-

yr]

0 for some k> 1,.r>1. (3.11) Asis wellknown,

R_=a subdirect sum ofsubdirectlyirreduciblerings

R, (i

E

F).

(3.12) Wenowtakeacloser lookatthe structureofeachof thesesubdirectsummands _Ri,withalleyetowards

provingtheircommutativity.

CASE1:

R

doesnothaveanidentity.

Lett:R

--

R be thenaturalhomomorphismofR onto Ri,and let t:x x Let N and Crdenote

thesetof nilpotents and thecenterofRi,respectively. Weclaimthat

R c

N

k.)C_i. (3.13)

Suppose

not. Let x E_Ri, x Ni, x Ci, and let o:x xi,

(x

E

R).

Then, clearly, x N and

x C,andhenceby(1.1),

x x Nforsomepositive integers n and m,n,m. Thisimplies(seetheproofofLemma (ii)) that

Byhypothesis(i), eisacentralidempotent,and hence x

xqe,

e e C.

Thisimplies,in

R,,

that

Xl

x,qe,,

e,

e,

C,.

(3.14)

Since

e,

is acentralidempotentinthesubdirectlyirreduciblering

R,,

therefore e, 0 (recallthat

R,

doesnothave anidentity),andhenceby(3.14),

x,

0, a contradiction, since

x,

isnotnilpotent. This contradictionproves(3.13).

Returningto(3.11),wesee that

[x,-x,y,-y]=O;

k>l,r>l; x,,y,

R,

(arbitrary). (3.15)

Now,bya trivialminimalityargument,it isreadilyverifiedtht(3.15)implies:

[a

i,b

i]

0 for allnilpotents ai,

b,

in

R,;

(i.e.,

N,

iscommutative). (3.16)

Combining(31.13)and(3.16),weseetha

R

is commutative.

CASE2: R hasanidentity.

Since the homomorphic image of a generalized periodic ring is also generalized periodic, it

follows thatR is commutative,by Corollary4.

Sinceeach R inthe subdirect sumrepresentation (3.12)is commutative, thereforethe ground ringRitself isalso commutative, and thetheoremisproved.

COLLARY6.

Any

generalized periodic ringwithcentral idempotents and commuting nilpotents

is commutative.

REFERENCES

HERSTEIN, I. N., Ageneralizationof atheoremofJacobonIII, Amer. J.Math. 75(1953), 105-111.