FINITE COMPLETELY PRIMARY RINGS IN WHICH THE PRODUCT OF ANY TWO ZERO DIVISORS OF A RING IS IN ITS COEFFICIENT SUBRING

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FINITE COMPLETELY PRIMARY RINGS IN WHICH THE PRODUCT OF ANY TWO ZERO DIVISORS OF A RING IS IN ITS COEFFICIENT SUBRING

YOUSIFALKHAMEES

Department

of Mathematics

Kng

Saud University P.O. Box2455 Ryadh 1145l, Saudi Arabia

(Received June 6, 1990 and in revised form August 30, 1993)

ABSTRACT.

According to general terminology, a ring

R

is completely primary if its set ofzero divisors fo.msan dcal.Let R bca finitecompletelyprimary ring.It iseasy toestablish thatJisthe uniquemaximal ideal ofR and R hasacoefficientsubringS (i.e. R/J isomorphictoS/pS)which is a Galois ring. In this paperwegive the construction offinitecompletely primary rings in which the productof anytwozerodivisors is in Sanddetermine their enumeration.Wealso show that finiterings in whichtheproduct^ofanytwozero divisorsis apowerof a fixedprime parecompletelyprimarytings witheitherJ"=0or theircoefficientsubringis

n

^with^{n=2 or}3. A specialcaseof these ringsisthe class of finiterings,studiedin[2],inwhichtheproductofany^twozerodivisorsis zero.

KEY

WORDS AND PHRASES.Finitecompletely primaryring. Galoisring.

1992AMS SUBJECTCLASSIFICATION CODES 16A ^10,16A^44,16A48.

1.

INTRODUCTION.

Alltingsconsideredin thispaperareassociative withidentity.Let Rbeafinitecompletelyprimaryring.

Itiseasy^tosee (cf. [5]) thatIRl=p’, Ul=pm-’", and the characteristic of

R

is p",for someprime pand positive integersm,nand withl<n<m. Ifn=m,thenRisof thelbrm

n[x]/(g)

and

R=Zpn[a],

where

Zn

istheringof integers modulop",gismonicpolynomialover

Zn

^and

irreducible

modulopand aisan element ofRof multiplicative orderp’-1.InthiscaseAut R,the automorphismgroupofR,is cyclic and isof order r. These rings areuniquelydeterminedby thetripletp,n, theyarecalled Galoisringsand aredenotedby GR(p",r).

Let R

beafinitecompletely primary ring. Itisalreadyknownthatany^twocoefficientsubringsofRare conjugate(cf. [4]).Also ifSis acoefficientsubringofR;thenthereexist rt,

min

^J^and^o’,

m

ⁱⁿ

Aut Ssuch that

R=S Srt

^(as

^S- modules) and

rtr

=r

rt

=1

for all in Sand for all i=l m.(This resultis a directconsequenceof theorems 2-2 and 2-4in[6]).

Moreover

the automorphismso, o,areuniquely determined by

R

andS (cf. [2]).Thus we call

, ,

the associated automorphisms ofRandthe automorphism

r,

iscalled theautomorphism associated with

.

^Throughout^this^paper, ^for^agiven finitecompletelyprimarynngR, we denoteby

T,

thesetof all(S,rt, ,)whichcome from the abovedescription. Inaddition,let F=R/J, and letF"and

G

^denote

themultiplicative group^ofunitsofF andRrespectively.

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CONSTRUCTION A" Let SbcaGaloisr=ng of the formGR(p",r)andFbeS/pS.^Also^assume^that^s,^t, w.m arcnon-negative integers such that^m=s+t+wandsupposethat is aninjcctive function from {s+l,

s+t} to{s+! m}.⁽⁾ⁿthe additivegroup R=SP’,^{define the}multiplicationas follows"

(r

_o _1’ _rn

)(So,

^S_1’

^s

_rn

(roSo+pn-

^U

^s

,=1

O

_{n_} _{s_..+} ^O ⁰

+P 2.r.s. ^,r ^{s +r} ^s ^{,r s}

^/r

^s

l__S+l o o o rn rn o

whereu,arcelements of F’,o,automorphisms ofFsuch that ,2=idF lot" alli=l sand

f,l=,"

for all

=s+ s+tandr" istheimage of under the canonicalhomomorphism fromStoF.

Itcan hceasilyverifiedthatRis aring andit iscommutativeifandonlyifo’,=id for allⁱ⁼ ^m.

THEOREM 1-LetRbeafinitecompletelyprimary ring. Then theproductof any twozero divisors is anclement ofitscoefficientsubring Sifand ifit isone of theringsgivenbyconstructionA.

PROOF: Let R be a finitecompletely primary ringwith .Fcontained in S and(S,

:/

ft.,’) be an clement of

T,.

SinceSS:,’=()andtheproductofanytwozerodivisors is inS, prt,’=0for all i=l m.

But

n:,’rt,’

is anclement of

pS;

thus

’rt,’

^isan element of

p""S

for all i,j=! ^m. Suppose

n:,’rt,’,’n,’

^are

non-zero elements of

pS

withj:k. Then

rt,’:,’S=K’rt,’S=p-’S

^{and we} get

’rt,’=rt,’’

^where ^ot ^{is an}

clement of<a>. Thus

rt,’-rt’c

^{is an}êlementôfânn

^rt,’

^andsubsequentlyit iscontainedin

pS :m Srt’ _h.

h=l,h l,k

This implies that

rt,’

^{is an}element of

pS(

_h=l,h ^m

Srt’h,

whichcontradictsthe assumption that(S,

rt,’,

rt,’)isanelement of

T,.

Therefore for all i=l m, either

r’rt,’

^is^zero^for^allj=l mor

r’:,’

isnon-zeroforonlyonej=l m.Similarly,weprovethat for alli= m,either

t,’r’

^iszero for all j= mor

rq’:,’

^is^non-zeroforexactlyonej. Assumewis the numberof

’

^{such that}

^n,’rq’

îs^zero^{for all j=} ^mând^%^{is the}^numberô,fôther

^rt,’.

^Let^us^reindex

n:,’, ...m:m’

ⁱⁿ^such^a^waythat for each i= kthere existsonlyonej= mwith

:,’r’=p"-’a,,

,where is anelement of<a>,and let f be the function from

}

to

m}

determinedby f(i)=j. Clearly

is injective. Also, for alli= m

n-

o,

(f(,) n-

o,

of(,)

p ao,f(i =/l:i/l:f(i) â â 71:/t:f(i) ^p â if(t)’

whichimplies that

cy,,=o’,-’

for alli= k.Letsbe the number of in

k

such that f(i)=i and be

,-s.

Wereindex

rt,’, rt’

^{such that}^f(i)=i^{for all}ⁱ⁼ ^and^suppose^o,l,l=u,^{for all}ⁱ⁼ ^s.^Put

^r,.,,=r’

for all i= sand

rt=rt’

for all i=s+l m,where if e is in the image of f,say e=f(i),then

a

=Fl(

e .=

fh-

^(,)

^fh(i

)g(h), where g(h)= (- 1) ^J+h+l

j-1

I-]c; and =1 otherwise.

d:h fO) ^e

Itiseasytoseethat(S.rt, rtm)is anelementof

T,

with

rt,rt,,,=lY"

for all i=s+l

..

^Now^it ^follows

thatRis isomorphictooneof the rings givenbyconstructionA.

The converseiseasy^tocheck.

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3.

FINITE

RINGS IN

WHICH THE PRODUCT OF

ANY TWO ZERO

DIVISORS

IS

A

POWER OF

A FIXED PRIME.

LEMMA

1"

Let R

be a finitering ofcharacteristic

p"

in whichtheproductofanytwozero divisors is a powerofp.Then

R

iscompletely primary.

PROOF:

Let x and y be zero divisors in R. To show that x+y is a zero divisor, we can use the distributivepropertiestowrite(x+y) as a sumofproducts,eachcontaining 2nfactors(whichare

x’s

or y’s). Sinceeach xyoryx is of theform

p,,

each of the summands of(x+y) is productof the form

p’p...pn=0.

Thereforex+y^{is zero}divisorand hence

R

iscompletely primary.

PROPOSITION 1’ Let R be a finite ring of characteristic

p"

inwhich the product ^ofany two zero divisors isapowerofp.ThenRiscompletely primarywith eitherJ-’=0orthe coefficientsubringofRis

Z.n,

where n=2,3.

PROOF:

Suppose

JZ-4:0; then there exist x,yin with

xy=p0.

SinceforanyunitoinR,otxis a zero divisor,we have03tx)y=p.".

On

theother hand,xy=p impliesthat^ct,xy=otp and so

otp-p

". Without loss ofgenerality,we can assume

la>__X

anddeduce that

p(ot-p"-)--0.

Since

p-0,

we have ot-p^"-xisanelement of

J.

If

ta

^this^would împly^thatôtîs ân^{element of}^J^which îs^notpossible; hence

la=k

ând^c îsân

element of l+J.

However

otis an arbitrary unit and therefore

G,=

l+J. Since

R=G,J

(disjointunion), wehave

IRI IG,I+IJI

I1+Jl+lJI 21JI

Thus2divides

IRI

andconsequentlychar

R

is2". Ifn>4,then 2,6 are zero divisors of

R

with(2)(6)=12 which isnotapower of 2.Also n= impliesthatJ2--0.Thus n=2,3.Let S=7_an[a]be a coefficientsubring ofR, where a is an elementof

R

ofmultiplicativeorder2’-1and letx,ybe elementsof

J

with

xy=2X0.

But (ax)y=2" implies a2-2"and hence a= 1. Thus the coefficientsubringof

R

is

Zn

with n=2,3.

4.

THE

ENUMERATION.

NOTATIONS:

Retaining the abovenotations,assume k is the number of elements in {s+t+l

m}

whicharenotintheimageoff.

Let

all thet,in which isnotintheimageoffberenamedas0,

0k

and assume

x,

% are therespective automorphismsassociated with them.Thus we

suppose

that(S,n,,

n, 0, 00

^{is an}^{element of}

TR

^and

,

^.,-k,

x,

"q,are theautomophismsassociated with

n,

r.,.,

0. 0k

respectively.

We

call

(p,

^{n, r,}^{s, t,}^{k, m,}f)the invariants ofR.

In

whatfollowsweshall use these notations.

PROPOSITION

2:

Let R

be a finitecompletely primary ring ^{in which}theproduct^ofanytwo zero divisors is anelement of its coefficient subring. Then(S,

n,’, r_’,

0,’,

0k’

^{is an}^{element of}

TR

^if^and

onlyif

0

=,la.O +p o.

= ^,

(after ^possi ^{bl e} ^{rei ndexi} ng), (after possi bl e rei ndexi ng ).

where

.,

^areelements of

F"

and

,,, ,,

la, areelements of

F

such that

,

^is^{zero if}

,

^is^notthe trivial automorphismand is zero if

x,

isnotthe trivialautomorphism.

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PROOF:Using the fact that

K’a=a , _K’,

wededuce that for alli= m-k,wehave

where

)%,,,

^and

,

^areelements ofF such that

,

^{is zero if}^o’,^is^notthe trivial automorphism.

For

all i= s+t,lann

,, I=IJI/tY

^{and so}

rt,’r,=0

^forall but onej, say j=h.Thus

’rt,,

îs^{a non-zero}êlementôf

p"-’S, ’r=0

^{for all}^j#f(i)^and

o,=(of,)"=o,.

Thus

,=0

^for^all exceptj=h. Letusput

),,,--.k,

andredenotc ft,’by

r’.

^Therefore

Wecanprovetherestof the propositionby using^asimilar argument.

THEOREM

2: Let R,R’ be finite completely primary rings constructed over the same coefficient subring

S

andhavingthe same associatedautomorphisms.

Suppose

that(J(R)) and(J(R’)) are contained in

S

andR,R’have the same invariantsp,n, r, s,t,k,m,f. Alsosuppose that(S,

:,’, rr.’,

0,’,

0)

is anelement of

TR.

^with

^rt’-" =p"-’v,

for all i=l s. ThenRisisomorphic to

R’

if andonlyif there exist isomorphisms

,

^from

^SSr,

^to

SS’

(after possible reindexing)for alli=l m-k such that

where

k,

^areelements ofF"such that

and

’hf(h)

h

^=I

for all i= s and h=s+l s+t,o<j<r.

PROOF: Let be an isomorphism fromR ^toR’. Then

(S)

^is^acoefficient subring ofR’and hence there exists a unit x in

R’

such that

(S)=xSx".

Let be thecompositionof theconjugation byxand

.

Clearly is anisomorphismfrom

R

to

R’

whichsends

S

toitselfand thus(S,

(n) (r), (0) (00)

^{is an}elementof

T,,.

Therefore for all i=l m-k

pn-

where

2q

are elements of

F

and

,,,,are

elements of

F

such that

,

^is ^{zero if}

^o,

^is ^not the trivial automorphism.

For

all i= s

n- U

pl pn-

P ((U,) (l)(/t:) ((l)(/1:,)) =(,i/l:, )2 pn-1, ’i"

Thus

, o,

Up

V" or some O<j

<r.

Also for all i=s+l s+t

0(/1: 2) (0(=)) =(.,/1:, )2 , /1:.

²

=pn-1, ^,(,

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Itiseasytosee that, for alli=l s+t,themappings

,

^from^S)S ^to

SSr’

determined by

,()=Lrt,’

areisomorphisms.

Conversely,let

,

^{be the}isomorphismsfrom

SSn:,

to

SS:,’

defined in the statementof the theorem, wherei= m-k.Itiseasytocheck that themapping determinedby

ro+,r,n

⁺

Er,o,)= ro+ ,Y_,r, ^,,(,)+ ,Y_,r,O’

isanisomorphismfrom

R

to

R’.

NOTATIONS:

Let

R

be a finitecompletelyprimaryringin which theproduct ofanytwozerodivisors isin itscoefficientsubringand letp,n, r,s,t,k,m, be invariants ofR. Assume

p

^isthe permutationon the maximal subsetof{s+l

s+t}

which isstable under and c is the number ofcyclesof

p.

Finally, let

a a

^p

for all s,

and

N,

be the number ofmutually non-isomorphic ringsof the form

S(Srt,

^withthe same associated automorphisms

,,

^where

rt,’-=p"-’u,.

Thenfrom theorem 2 in[31,^{we have}

2 pr/Z+l

if p is even and . ^is ^the ^trivial automorphism, if pisoddand(r is thetrivial automorphism, if . ^is ^not the trivial automorphism.

THEOREM 3:The number ofmutually non-isomorphicfinite

completely

primary ringsin which the productofanytwozerodivisorsis inits coefficient subring,havingthe same invariantsp,n,r,s, t, k, m, fand with the same associated automorphismsis

(pr_ 2)t’c 12I

I=|

^N..

PROOF:

If u,,v,areelements ofF’,defineu,_v,ifandonlyif

r, p +1 Up

V"

forall i=l s,where0_<j<r.

By

using similarmethod as in the proofof theorem 2 in [3],one can deduce that the number ofequivalenceclassesof this equivalent relationis

N,.

Define

t._rt,’

ifand only if

:=,n:,’

for alli=s+ s+t,where isanelement of

F"

such that

,fo=l.

^Letn, be the number of the equivalenceclasses of thisequivalentrelation.Thenn,= if isnotintheimageoffand

n,=p’-2

if is in the imageoff.

But

whenfrestrictedto

{s+l s+t}

the numberofelements inthe imageoffis t-c.

Thus

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I=S+l

fin

(pr- 2)t-c.

Inview ofthe last theoremthe required numberis

=1 =s+l =1

COROLLARY:

The finiteringof characteristic

pn

ⁱⁿ which theproductofanytwozero divisors ^tsa powerofpiscompletelydetermined

by

its associatedautomorphismsand its invariants.

REMARK:

Let

R

be a finiteringwhichhasap-ring asitscoefficientsubringand theproductofany

twozero divisorsofRis in itscoefficient subring.

By

using similar argument as in theproofof lemma 1, necanprovethat

R

iscompletelyprimary. Thus the construction and the enumeration of suchrings^is

determined.

ACKNOWLEDEMENT:

The author would liketothank

B.

Corbas for hissuggestionswhichenabled the authortomake someimprovementsinthecontentsof the

paper.

REFERENCES

1. Y. A1Khamees,

On

thestructureof finitecompletely primaryrings,J. Coll. Sci., King Saud Uni.

13(1)(1982),149-153.

2. Finitetingsin whichthemultiplicationofanytwo zerodivisors iszero,Arch. Math.

Vol.37(1981), 144-149.

3.

Near

Galoisrings, Proceedings of theconferenceonAlgebraand

Geometry,

Kuwait (1981), 1-6.

4. W. E.Clark,Acoefficientringfor finite non-commutativetings, Proc. Amer.Math.

Soc.

33(1)(1972), 25-27.

5. R. Raghavendran,Finite associativerings, CompositioMath.21(2)(1969L 195-229.

6. B.R.Wirt, Finitenon-commutativelocal rings, Ph.D.Thesis,Uni. ofOklahoma,

(1972).

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This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

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FINITE COMPLETELY PRIMARY RINGS IN WHICH THE PRODUCT OF ANY TWO ZERO DIVISORS OF A RING IS IN ITS COEFFICIENT SUBRING