On the compactness of the structure space of a ring

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ON THE COMPACTNESS OF THE STRUCTURE SPACE OF A RING

~ (Wuhan Lee)

1. INTRODUCTION Jacobson (l] has shown that a topology may bedefined on the set SeA) . of primitive ideals of any non-radical ring A With this topology SCA) is called the STRUCT·

URE SPACE of A The topology is given by defining closure: If T=(P} is a set of primitive ideals then T is the set ofprimitive ideals which contain

0,.=n^{{PIPE T}.}

It is well known that if A has an identity el·

ement. then SCA) is compact (2. pp. 208).

Moreover M. Schreiber (3) has recently observed that if every two-sided ideal ofA is finitely generated. then SCA) is again compact. However, since the condition that A has an identity element neither implies nor is implied by the condition that every ideal of A be finitely generated. it is clear that neither of these conditions is necessary in order that SCA) be compact. R. L. Blair and L.C. Eggan (4) investigated this situation thro- ughlyandfoundacondition that is both necessary and sufficient for the compactness of SCA) as a consequence of a general lattice-theoretic result.

They also obtained a remarkable result for aclass of rings consisting of those ringsA such that no non-zero homomorphic image of A isaradical ring, stating that the structure space of such ring is compact if and only if A is generated. as an ideal.

by a finite number of elements. Recently a direct proof wasgivenbyTaikyunKwun (5)usingopen sets instead of closure. In view of the fact that the Jacobson radical of an arbitrary ring pl2_Ys an important role in the structure theory of rings.

Presented at the Yonsei Simposium, 9 Oct 1962, received 2 Mar. 1964

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the author tried to find a relationship betweell the notion of radical and the compactness of a structure space. As a result he found that for a certain class of rings the modularity of the radical is both necessary and sufficient for compactness of SCA). and for anotherclassofrings the condition is that A is generated as an ideal by a fmite number of elements. Section2is entirely due to M. Schreiber (4)and the author has found his method very useful in searching for a link between the compactness and theradical. Section 3 is concerned with properties of the modular ideal which will be' used in the proofs of the main theorems. Section 4- and 5 contain the main.

theorems.

2. OPEN BASIS OF THE TOPOLOOY. For each xEA write (x) for the principal two-sided ideal generated by x. and let

Ux= {PIP=l2(x). for all PESeA)}.

PROPOSITION 1. lUx}1UA is an open basis ot the topology.

PROOF. Since the set {PIP.::l(x)} is clearly closed, its complement Ux is open. Let U be an open subset of SCA), and let F=UC (the complement of U). Now suppose PEU. Since F=F

= {F'Ip/::2Dr>} and PG F. we have P Dr>.Hence there exists an element a in A such that a be- longs to D_F but not to p. so that P (a). that is, PE U... Suppose pIEU... then pI=I=-(a) and

P~F.so that F'1$F. or p/€U. HencePE UaCU.

PROPOSITION 2. If a ring A has an identity then SeA) is compact.

PROOF. We prove that any basic open cover has a finite subcover. By Proposition 1. the col- lection

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Write

and

A(l-e)= {x-xerxEA}, A(l-eD= {x-xCII xEA}.

It follows A(l-e)=A(l-~)(l-e,)

lUx}^X_A is an open cover of SeA). Then

SeA)=UlUx} x_A

= {P13 ^).Isuch that P:;J2Ca_v)}

=^{PlI;:j~I:a.A Ca)}.

(AfR)f(PfR)t;;;tAfP.

Hence PfR is a primitive ideal in AfR for AlP is primitive ring. Conversely any primitive ideal in AfR is of the form PjR, where P is a

=(x-xe2)-(x-xe2)el•

On the other hand, A(l-e2)cA. and this implies

A(l-e2)(l-e,)k:ACl-e,).

But A(l-eJ)c:P. Hence we have A(l-e)cP.

Since A(l-e2)k:P'. we have A(l-e2)(l-el)

cP/(l-ea. Therefore

A(l-e)c;P' and A(l-e) cPnP'.

A similar argument implies (l-e)Acpnp'·

Hence P n P' is modular.

PROPOSITION 4. Every two-sided modular ideal can be imbedded in a primitive ideal.

PROOF. Let (I:A) denote the set of element a of a ring A such that AacI where I is a modular maximal right ideal. Since

(I:A)=CO:M)

where M is an irreducible A-module (2, Pro- position 2. pp. 6). it follows that (1:A) is primitive ideal. Now let B be a modular two-sided. ideal. Since it is well known that every modular right ideal can be imbedded ina modular maximal right ideal. regarding B asamodular right ideal. we can put Bb!. But· Abc:B for every element b of B. Hence Bc(l:A).

4. COMPACTNFSS AND THE MODULARITY OF THE RADICAL. By Proposition 5 below.

the modularity of the radical of a ring A is sufficient for the compactness of SeA). But it is not necessary for the same reason stated. in the INTRODUCTION. An effort was done in searching a class of rings for which the compactness of SCA) necessitate the moduIarity of the radicals.

PROPOSITION 5. H the radical R of a ring A is modular. then SCA) is compact.

PROOF: Let e be the identity moduloR. Then AfR is clearly aring with an identity. Since AfR has an identity. the structure space SCAfR) is compact. Now consider a primitive ideal P. Then we have

x-xe=x-x(e2 0 eJ)

=x-xC~+eJ-~e,)

=x-xez-xeJ+x~eJ

I=E__

A Ca).

In a ring with an identity every two-sided ideal can be imbcdded in a primitive ideal (6). But

{PjF=tI}=SCA)

exhausts aB primitive ideals. Hence I=A, so that the identity element I is in 1 Hence there exist b,•...• bn in (a)+··+(b) sU:ch that the identity 1= b, + ... + bn• so that

A=(a)+···+(b)=l

But this means that there exists a finite subset E={a.···.b} of A such that

SCA)= {PIF$E""A(a)}

=U(U"}"'E' This proves the Proposition.

3. MODULAR IDEALS.

DEFINITION. A two-sided ideal P is called modular if and only if there exists an element e in A such that for all a of A, a-ea, a-ae E P.

The element e is called an identity modulo P.

Evidently. if a two-sided ideal P of ring A is modular with an identity e modulo P, then AlP

is a ring with an identity e+P.

PROPOSITION 3. An intersection of finite number of modular two-sided ideals is modular.

PROOF. Let P and pI be modular two-sided ideals of a ring A. and let eJ and ~ be the identities modulo P and pI respectively. Then it suffices to show that the intersection of P and

pI is modular. Now put

e=~ ⁰ eJ=~+e,-~el

since

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primitive ideal. Since the correspondence P-4PjR

preserves arbitrary intersection. itfollows that it is a homeomorpism of SCA) onto the structure space S(AjR) of AIR. Therefore SeA) is com- pact

Now we consider a class of ring with the property that

(C) For every D. of fU.} ....A. Du.is modular.

For such a ring A. a result can be obtained as follows:

PROPOSITION 6. For a ring A satisfying the condition CC). if SCA) is compact, then the radical R of A is modular.

PROOF: It suffices to show that any basic open cover has a finite subcover. Since {Dx } xeA

is an open cover. there exists a subcover {D.}aeE

where E is a finite subset of A. Then

Du{u.l=Ds{A)=R. (R: the radical of A) since

But

Du{u.}:::2Du.nDubn... nDue.

where a.b, ..·,c^€E.

Hence

R.:::2DuanDubn... nDue.

where a.b, "',cc: E.

By hypothesis. each {Dua}. ac:E. is modular.

Then it follows that the radical R of A is modular by Proposition 3. This proves the Propos- ition.

Combining proposition 5 and 6, we obtain the following:

TBmREM. 1. Let A be a ring with a property that each Du. is modular. Then the structure space of A is compact if and only if the radical R of A is modular.

5. COMPACTNmiS AND MODULARITY OF P- IUNCIPAL TWo-sIDED IDEALS. Itispointed out explicitly in (5) that the condition that A is generated. as an ideal. by a finite number of elements is sufficient for the comactness of SCA) regardless of the type of A. We consider a class of rings with the property that

(C') Every principal two-sided ideal is modular.

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PROPOSITION 7. For a rir.g A satisfying the condition CC'), if SCA) IScomp3.ct, then the ring A is generated. as an ideal. by a finit::l number of elements.

PROOF: Consider the basic open cover fUll]xfA

Then we have a finite subcover {D.}a€E

where E is a finite subset of A. and SCA)=U(D.I.fiE

= {P[P=.t'X,.,E(a)).

Write

B=E.m(a).

Since B contains a principal ideal. it is modular. Now suppose B is proper Ideal of A. By proposition 4, B can be imbedded in a primitve ideal This is a contradiction to th3 fact that

{PIP$B}=SCA)

exhausts all primitive ideals. Hence B=A.

We state this fact in the following fonn:

THEOREM: 2. Let A be a ring with the property that every principal two-sided ideal is modular. Then the'structure space of A is compact if and only if the ring A is generated. as an ideal by a finite number of elements.

REJIlAKKS: Considera ring A with a pr0-

perty that

(C") No non-zero homomorphic ^J:JDagCof A is a radical ring.

R.L. Blairand L.C. Eggan have proved that.

for sucha ring, the structure space is compact ifand only ifA is generated as an ideal by a finite number of elements. It would be interesting to clarify the relations among the classes of rin- gssatisfying codition CC). CC') and CC").

REFERENCES

1. N. Jacobson. A topology 01 thesetof/Jrimitive ideals in an arlJitrary ring. proc. Nat. Acad.

Set. U.S.A. tool. 31 (1945) pp. 3 3-338.

2. N. Jacobson. Structure of Rings. Amer. Math, Soc. Colloquiums. Publications. vol. 37, Provid- er.ce, 1956.

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3. M Schreiber. ComPact'lless 01 the structure sp- ace 01 a ring. Proc. Amer. Moth. Soc. vol. 8 (1957) pp. 684-685.

4. R.L. Blairand L.C. Eggan, On the compactness 01 the structure space of a ring. Proc. Amer

Math. Soc. vol. 11 (1960) pp. 876-879.

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'%4.(H.M Wcd.derburn~ Lectures on matrices.

Amer. Math. Soc. Colloq. Pub!. Vol 17. 1934~

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:fll~oJl M Marcus~ N.A Khan~ (Canad.

J. Math. 12(1960). 259-277) fil:lk6*-i- Ail- . AX-XA~ RI~ol'i!'8;fAJ- X~ XB-BX$} 1iJ~

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5. Taikyun Kwun. A direct proof oj theorem oj BlairandEggan and some results (to appear).

6. N. Jacobson. The radical and semi-simPlicity lor arbitrary rh'g. Amer. j. Math. trot. 65

(1945) pp. 300-320.

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