I-RINGS AND S-RINGS

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 4 (1997) _825-827

825

SUBRINGS

OF

I-RINGS AND S-RINGS

MAMADOUSANGHARE

Drpartement

deMathrmatiquesetInformatiques Facult6 desSciencesetTechniques

UCAD DAKAR (SENEGAL) e-mail sanghare@ucad,^refer,sn

(ReceivedMay6,1993and in revised formFebruary 13,1997)

ABSTRACT. Let

R

bea non-commutative associativeringwithunity1

:f:

^0,aleft R-moduleis saidto satisfy property (I) (resp.

(S))

^if every injective (resp. surjective) endomorphism of M is an automorphism ofM. ItiswellknownthateveryArtinian(resp. Noetherian)module satisfies property(I) (resp. (S))and that the converse isnot true. Aring

R

iscalled a leftI-ring (resp. S-ring)ifeveryleft R-module withproperty(I)(resp (S))is Artinian(resp.Noetherian). Itisknown that asubringBof a left I-ring (resp. S-ring)

R

isnotingeneralaleft I-ring (resp. S-ring)^evenifRis afinitelygenerated B-module, for examplethering

M3 (K)

^of3x 3 matrices over a fieldK isaleft I-ring (resp S-ring), whereas itssubring

B=

a 0

7 0

which is a commutativeringwith anon-principal Jacobsonradical

J=K. 1 0

+K.

0 0 0

00 100

isnotanI-ring (resp. S-ring)(see [4],theorem8). WerecallthatcommutativeI-rings (resp S-tings)are characterized as those whose modules are a direct sum of cyclic modules, these tings are exactly commutative, Artinian, principalidealrings(see[1]). Someclassesofnon-’commutativeI-tingsandS- tings havebeen studied in [2] and

[3].

Aring

R

is offinite representationtype if it isleftandfight Artinian and has (up to isomorphism) only a finitenumber of finitely generated indecomposable left modules. In the case of commutativetingsorfinite-dimensional algebrasover analgebraically closed field,theclasses of left I-tings, left S-tings and tings offiniterepresentationtype are identical(see [1]and

[4])

^Aring

R

is said tobe a ringwith polynomial identity (P. I-ring) ifthere exists apolynomial

f(X1,X2,...,X,),

^r

>

2, inthenon-commutingindeterminatesX1,X2,...,X, overthecenter ,7ofR suchthat oneof themonomialsof

f

of highest total degreehas coefficient 1,and

f(al,

^a2,

a)

⁰

for allal,a2,

a

inR. Throughoutthispaperallringsconsidered are associativeringswith unity, and byamodule

M

over aring

R

wealwaysunderstandaunitary left R-module. Weuse

MR

^toemphasize

that

M

is aunitary fight R-module.

KEY WORDS AND PHRASES: Left I-ring, left S-ring, ringwithpolynomial idemity, ring offinite representation type.

1991AMSSUBJECTCLASSIFICATION CODES: 16D70, 16P10, 16L60.

826 M. SANOHARE 1. THE MAIN RESULT

THEOREM. LetRbealeftI-ring (resp. S-ring),and

B

be asub-ring of

R

contained in thecenter

Z

of

R

Supposethat

R

is afinitelygeneratedfiatB-module Then

B

isanI-ring (resp S-ring) To provethistheoremweneedsomeresults.

Itiseasytosee that

LEMMA

^1.

Every

homomorphic image ofaleft I-ring (resp S-ring)is aleft I-ring (resp S-ring) LEMMA 2. Let

Px

^and

P2

^{be twoprimeidealsof aringR. If}

P1

^{is not}contained in

P2

^then

Hom(R/P, R/P2) {0}

PROOF. Let

f- R/P1 R/P2

^{be an} R-homomorphism, and set

f ⁽¹ + P1 +

^{P2, where}

ER. Letx E

P1 \P2,

^{and letr}be any elementinR. Wehave

P2 ^f(xr + PI)

xrt

+

^{P_ Thus}

xRt

P2.

^Since

P2

^{isprime,wehave 6P2,andhence}

f

^0.

LEMMA

3. Let

R

^beaprime ringwithpolynomial identity. If

R

is aleft I-ring(resp.S-ring), then

R

issimpleArtinian.

PROOF. Let

R’

bethe totalring of fractions of

R

[5]. Itisknown that

R’

issimpleArtinian[5],so theR-module

R’

satisfies(I)(resp.

(S)).

Since

R

is aleft I-ring (resp. S-ring), then

R’

is an Artinian (resp. Noetherian) R-module and hence

R

R.

LEMMA

4. Let

R

^beasemi-prime ringwithpolynomial identity. If

R

isaleft I-ring(resp S-ring), then

R

issemi-simple^Artinian.

PROOF. Let

(Pt)te.

^{be a}family pairwisedistinct minimalprimeidealsof

R

such that

teL

By Lemma the quotient tingsR/Pt(g

L)

areleft I-tings (resp. S-rings)withpolynomialidentity Thenitfollows from Lemma3 that the tings

R/Pe( L)

are simpleArtinian, so theleftR-modules

R/Pt(g L)

satisfy

(I)

(resp

(S)).

FollowingLernma1,

HomR(R/Pt, RIPe) {0}

^for

",

sothe leftR-moduleM

tet.R/Pt

^satisfies(I) (resp. (S)). Since

R

is aleft I-ring (resp. S-ring), thenMis Artinian. ButR regardedasleft R-moduleisisomorphic^toasubmodule of the semi-simpleArtinianleft R-module

M,

hence

R

issemi-simpleA.,’tinian.

PROPOSITION5. Let

R

bearingwithpolynomial identity. If

R

is aleft S-ring (resp. I-ring), then

R

isleftArtinian.

PROOF. Supposethat

R

is aleft S-ring (resp. I-ring)thenthe quotient ring

R/rad(R),

^where

tad(R)

^is the prime radical of

R,

is a left S-ring (resp. I-ring), so, follqwing Lemma 4, the ring

R/rad(R)

issemi-simpleArtinian Thisfactimpliesthat

R

issemi-perfect and hence

tad(R) J(R),

where

J(R)

^is the Jacobson radical of R. Let e be a primitive idempotent ^of R. Since the endomorphism ringof theR-moduleReisisomorphictothelocal ring

ere

^witha nil Jacobson radical

eJ(R)e,

thenthe R-moduleRe satisfiesproperty

(I)

(resp (S)). Itfollowsthat theR-module

Re

^is

Noetherian (resp. Artinian). Since

R

regarded ^as R-module is direct sum of finitely many left R-modules of the form

Re,

whereeisaprimitive idempotent of

R,

then

R

is Noetherian. Let

P

now be aprime ideal of

R.

Since theprime ring

R/P

^{issimplein virtue}of Lemma 3, then

R

isleftArtinian.

PROOF OF

THE

MAINTHEOREM. Since

R

is afinitelygeneratedZ-module, then

R

is a ring withpolynomialidentity(see [6]). Soby Proposition 5

R

is aleftArtinianring. Thusby

[7]

the ring

B

is Artinian. Let el ,e, be primitive idempotents of

B

such that B

=e,

^{Bei For everyi,}

1

<_ <_

n,

B,

e,

Be,

isalocalArfinian ring. ToshowthatBis aleftI-ring(resp. S-ring)it isenough to show thatforevery i, 1

< _<

n,

B,

is aleft I-ring(resp. S-ring). Wehave

A

_$=IAi, where

A,

e,

Ae,,

1

_< _<

n. Byhypothesis theleftB-module

=1 ^{A, A}

is flat andfinitelygenerated,so the

Bi-module

A ^eAe, - ^eiAe

^(R)BB

^A

^(R),e,

^{Be A}

SUBRINGSOFI-RINGSAND S-RINGS ⁸²⁷ isalso flat and finitely generated Since

B,

isan Artinianlocal ring thentheB,-module

A,

isfaithfully flat(see[8]proposition1,p.44)

SupposenowthatB, isnotanI-ring(resp. S-ring) forsome i, 1

_< _<

n Thenby Proposition²of [2],there exists aB,-moduleMofinfinitelength such that, for every integern

_>

1, theB,-moduleM satisfies both properties(I) and (S) Following [$] (corollary 2, p. 107), the B,-moduleA, is afree module. Let

M’

M

(R)B,A,.

^Sincethe

Bi-module

^M isofinfinitelengthand

A,

is afaithfully flat Bi-module, then

M’

is anA,-moduleofinfinitelength. Onthe otherhand,since

A,

is afree B,-module, thereexists anintegers

_>

Isuch that

A, B.

^Wehave then theB,-moduleisomorphism

M’

M

(R)B,A M (R)B,Bt -

^M’.

Hencethe B,-module

M’

M satisfiesboth properties (I) and (S) and therefore

M’,

regarded as A,-module,satisfiesproperties(I)and(S) Thisfact impliesthatthe homomorphic image

A,

oftheleftI- ring (resp. S-ring)

A

is not aleft I-ring(resp.S-ring),in contradiction withLemma1.

COROLLARY. Let

R

bealeft I-ring(resp. S-ring). If

R

is afinitelygenerated flat moduleover itscenter

Z,

thenZis anI-ring (resp. S-ring).

Thefollowing example showsthattheconverseof thetheoremaboveisnot true" LetKbe a field The commutativeringA

K[X,Y]/(X2,XY,

^Y

2)

isnotanI-ring (resp. S-ring)because its Jacobson radical 3’ KX

+

^{KYis not}principal(see [1],theorem8). Ontheotherhand

K

isanI-ring (resp S-ring)and

A

is afinite-dimensionalK-vectorspace

ACKNOWLEDGEMENT. Theauthor wouldlike to thank thereferee forhisvaluable suggestionsand numerousvery useful remarks aboutthetext.

REFERENCES

KAIDI,

A.M.andSANGHARE, M., Unecaract6risation des anneauxartiniens/tid6auxprincipaux, Lec.NotesinMath.,Vol. 1328,Springer-Verlag,Berlin(1988),245-254.

[2] SANGHARE, M., Sur quelques classes d’armeauxli6es aulemme deFitting,Rend. Sen,. Math.

Padova,87(1992),29-37.

[3] SANGHARE,

M.,

OnS-duo-rings,Comm.

m

Algebra20(8) (1992),2153-2159

[4] SANGHARE,

M.,

Characterizationsofalgebraswhose moduleswithFitting’s propertyareoffinite length,Ext. Mat.7(2)(1992),1-2.

[5] POSNER,E.C., Primerings satisfying apolynomial identity,Proc. Amer.Math. Soc. 11 (1960), 180-183.

[6] RENAULT,G., Algdbrenon-commutative,Gauthier-Villars,Paris(1975).

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I,

Hermmm,Paris(1961).