3: 21 is a two-sided deal and if W has an identity, 3: 21 is the largest twosided

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1 VOL. 31, 1945 MA THEMA TICS: N. JACOBSON 333 lations when the number of new cases is less or only slightly greater than the number of new susceptibles during each incubation period. 'f Wilson, E. B., Bennett, C., Allen, M., and Worcester, J., Proc. Amer. Phil. Soc., 80, (1939). See particularly Appendix II, Localness of Measles, A TOPOLOGY FOR THE SET OF PRIMITIVE IDEALS IN AN ARBITRARY RING By N. JACOBSON DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY Communicated July 11, In a recent paper we have called a ring 2 primitive if W contains a maximal right ideal a whose quotient 3: W = 0.1 In general if 3 is any right ideal 3: 21 is the totality of elements b such that xb e a for all x in W. 3: 21 is a two-sided deal and if W has an identity, 3: 21 is the largest twosided ideal of 21 contained in 3. The primitive rings appear to play the same r6le in the general structure theory of rings that is played by simple rings in the classical theory of rings that satisfy the descending chain condition for one-sided ideals. Corresponding to the Wedderburn-Artin structure theorem on simple rings satisfying the descending chain condition we have the theorem that if W is a primitive ring 5 0,?1 is isomorphic to a dense ring of linear transformations in a suitable vector space over a division ring. We shall call a two-sided ideal 23 in 21 a primitive ideal if e0 $ 2 and 21- Q3 is a primitive ring. It is known that if 21 is not a radical ring then W contains primitive ideals. Moreover, in this case the intersection IIH of all the primitive ideals in W coincides with the radical T of W. In particular if 21 is semi-simple, I13 = 0. If W is a ring with an identity, 21 is not a ra'dical ring. Hence if E is any two-sided ideal 5 21 in 2, 2 -E contains a primitive ideal ez - (S. Since 21 - _3 - (21 - y) - ( -(), e is primitive in 2. Thus if 21 is a ring with an identity any two-sided ideal ( 5 2 can be imbedded in a primitive ideal. Any commutative primitive ring is a field. Consequently any primitive ideal in a commutative ring is maximal. On the other hand, in a noncommutative ring there may exist primitive ideals that are not maximal. For example, the ring 2 of all linear transformations in an infinite dimensional vector space, 9 over a division ring is primitive. Hence (0) is a primitive ideal. However, 2 contains as a proper two-sided ideal the set a of finite valued linear transformations in T.

2 334 MA THEMA TICS: N. JACOBSON PROC. N. A. S. A simple ring is either primitive or a radical ring. In particular any simple ring with an identity is primitive. These remarks imply that a maximal two-sided ideal Q3 such that f - e is not a radical ring is primitive. If g is a ring with an identity, any maximal two-sided ideal in S is primitive. In this note we shall define a topology for the set of primitive ideals of any ring. The space determined in this way appears to be an important invariant of the ring. We hope to discuss its r6le in the general structure theory in greater detail at a later date. 2. Let S be the set of primitive ideals in the ring 5(. S is vacuous if and only if l is a radical ring. If A is a non-vacuous subset of S we let Z)A denote the intersection of all the primitive ideals e3 e A. We now define the closure A of A to be the totality of primitive ideals z such that >- A. It is clear that 1. A> A. 2. A =A. We wish to prove next that 3. A VB = A V..2 Proof. LetQeA vib,saye A._ThenQe._ZA. Hence e >-ZA A ZB= )c where C = A v B. Thus e e A v B. Suppose next that fi e A v B. Then e3 > ZA) and Qe > ZB. We consider now the primitive ring W -. The two-sided ideals (ZA + Z) -e and (ZB + ) -e are $ O in -Z. Hence the product [(TZA + Z) - Z] [(ZB + Z) - ] 5 O. It follows thatz)az)b 5. Hence also Z)AA )B eand 3 so53e A v B. For the vacuous set w we define 4. C = w. The properties 1-4 show that S is a topological space relative to the closure operation A -) A. In the special case of a Boolean ring this topology is due to Stone.4 It has also been introduced by Gelfand and Silov in commutative normed rings.5 We shall call the topological space S the structure space of the ring W. If Q3 is a primitive ideal in 2X, e is a point in S. The closure of the set {d} is the totality of primitive ideals ( such that ( _ Q3. Hence if { } = 1tQ2} then Z, = Q2. This shows that S is a To-space. In general S is not a T1-space. For we have seen that there exist primitive rings 21 that are not simple. If W1 is a ring with an identity of this type and e is a proper two-sided ideal in 2, Z can be imbedded in a primitive ideal G. Then I(O) } contains ( 5 0. The subspace M of S of primitive ideals that are maximal is clearly a Ti-space. If 21 is commutative S = M is a Tl-space. However, even in this case S need not be a T2- (or Hausdorff) space. An example of a normed ring of this type has been given by Gelfand and Silov.5 A simpler one is the following:

3 VOL. 31, 1945 MA THEMA TICS: N. JACOBSON.335 Example. Let 21 = J the ring of integers. The primitive ideals are the prime ideals (p). Since the intersection of an infinite number of prime ideals is the 0-ideal, A = S for any infinite set A. If A is finite, A = A. Hence the open sets cw, 5 S are the complements of finite sets. Any two open sets $ w have a non-vacuous intersection and so the Hausdorff separation property does not hold. 3. Let 911 be an arbitrary two-sided ideal in 21 and let S1 denote the closed set in S consisting of the primitive ideals!3 of 2 that contain 21k. If e e S1 then e - 2f1 is a primitive ideal in 21-2f and any primitive ideal in is obtained in this way. The correspondence Q5-* 53-2f1 is (1-1) between the subspace S1 of S and the structure space T of the ring Since this correspondence preserves intersection it is a homeomorphism between S1 and T. Suppose in particular that 2h = T the radical of 21. Then every primitive ideal of 21 contains 9. Hence Si = S, and we see that the structure space of 21 is homeomorphic to the structure space of the semi-simple ring - 9?. We return to the general case in which 211 is arbitrary and we now consider the open set Si' of primitive ideals that do not contain 21k. Let e Si'. Then 9 A 21f is a two-sided ideal # 21f in 1i and 21f - (S A 21k) ((E + 2t1) -. The latter ring is a two-sided ideal in the primitive ring 21 - (S. Hence it is primitive.6 Thus ( A 2[1 is in the structure space U of the ring 21. Let A be a subset of Si' and let ( e A A S,' so that X is in the closure of A in the subspace Si'. If ZA is the intersection of the primitive ideals in A then ( > Z but Z > 21. Let B be the subset of U of ideals e8 A 21f where e e A. The intersection of all of these ideals is the ideal ZA A2A1. Since (( A 21k) > (ZAA 2f1), Z A 21 is in B. Hence the mapping that we have defined between Si' and the subspace of U is a continuous one. 4. Let {Faj be a set of closed sets in the space S. As before let Z'Fa denote the two-sided ideal of elements common to all the e e Fa. Suppose that the intersection. ]IFa,5w and let e be a point in this intersection. Then e > Z)Fa for all a. Hence e contains the two-sided ideal 2;ZFa generated by the ZFa' The converse follows by retracing the steps of this argument. We therefore have the LEMMA 1. If I Fa I is a set of closed sets in S, HFaF. O if and only if 2ZFa can be imbedded in a primitive ideal. If 21 is a ring with an identity any two-sided ideal 0 21 can be imbedded in a primitive ideal. Hence we have the Corollary. If 21 is a ring with an identity and (Fa} is a set of closed sets in S then HF,, w3 if and only if 2Z)Fa y 21. If 21 has an identity and :2Fa = 21, 1 = di dt where di e TDFi. Hence also 2i)Fj = 21. The corollary therefore shows that if IIF, = co

4 336 MA THEMA TICS: N. JACOBSON PROC. N. A. S. then there is a finite set of closed sets F1 such that HIF1 = co. We therefore have the following THEOREM 1. The structure space of a ring with an identity is bicompact. Suppose now that 21 is a semi-simple ring. Let W be decomposable as a direct sum 91 (D %2 of the two-sided ideals 21i $ 0. Let St be the closed subset of S consisting of the primitive ideals e3 containing the ideal 21. Since ( 212 a semi-simple ring, SI 0 w. Similarly S2 0 w. Also the semi-simplicity of 2-21i implies that 21 is the intersection of all the e in Si. Since = 21 the lemma implies that Si A 52 = co. Now let e3 be any primitive ideal in 21. We assert that either 53>2-1 or e For otherwise W2 + e3 > 5d. The ideals (21w + 13) - e3 are $ 0 in the primitive ring Since [(21, + 5) - ] [(21 + ) - ] = 0 this is impossible. We have therefore proved our assertion. Evidently it is equivalent to the relation Si v S2 = S. two components S, and S2. Conversely suppose that S = S, v S2 where the St are closed sets $ w such that Si A S2 =. We assume also that 2 has an identity. Let 21 = Zs. Then 21 + W2 =2. Since 21 is semi-simple W1 A %2 = S Thus S is disconnected into the 0. If21 = 0, S = Si = Si contrary to S1 # w and S2 $ w. A part of our result is the following THEOREM 2. If 2 is a semi-simple ring with an identity, 21 = 1I ff 2 where the W2 are two-sided ideals $ 0 if and only if S is not connected. 5. If S' is a completely regular bicompact space and 25 is the ring of realvalued (complex valued) continuous functions on S' then it has been shown by Gelfand and Silov that the structure space S of e5 is homeomorphic to S'.7 If S' is a bicompact totally disconnected space and Si is the ring of continuous functions on S' having values in the field of residues mod 2, then by a result of Stone's the structure space of e is homeomorphic to S'.8 We conclude this note by giving another example of this type based on an arbitrary totally disconnected bicompact space S' and an arbitrary division ring SU. We consider any decomposition of S' into a finite number of components (non-overlapping open and closed sets) S'i and we choose corresponding elements k1 e-'. We define a function f(x) by setting f(x1) = k1 for xi in S's. A function of this type will be called a finite decomposition function. The totality S of these functions is a ring under the ordinary operations of addition and multiplication. i contains the subring S of constant functions, isomorphic to S' and S is commutative if and only if,' is commutative. The constant 1 acts as an identity in 25. Since S' is totally disconnected, for any two points a d b in S' there is an f(x) e S such that f(a) 0 f(b). Let 2 be any subring of (S having this property and containing R. We shall sketch a proof of the fact that the space M of maximal two-sided ideals of 21 is homeomorphic to S'.

5 VOL. 31, 1945 MA THEMA TICS: N. JACOBSON 337 LEMMA 2. If F' is a closed subset of S' and a is a point i F' then there exists afunction.p(x) e 2X such that p(y) = Ofor all y e F' but (epa) $ 0. Our assumptions imply that for each y e F' there is an fy e a such that fv(y) = 0 butf,,(a) $ 0. The set Z'(fv) of zeros offy is open and the totality of these sets covers F'. Let Z'(fy,),..., Z'(fy) be a finite subset of these sets covering F'. Then so(x) = fvl(x)... f,im(x) has the required properties. If a e S' we let Q3a denote the totality of functions g(x) e 2t such that g(a) = 0. It is easy to see that 3a is a maximal two-sided ideal in 2t and that S- )3a " SR. Our conditions imply that if a $ b then e3a #-23b- LEMMA 3. If Z8 is a two-sided ideal $ W., then there exists a point a such that g(a) = Ofor all g e 3. Let Z'(f) be the set of zeros of f. Z'(f) is closed. If the lemma is false, the intersection 11Z'(f) for all f e e3 is vacuous. Hence there is a finite, number of functions fi,..., fr in e such that 1IZ'(f1) = w. Hence if a is any point of S' at least one of the functions fi does not vanish at a. ki)...., knii be the non-zero values taken on by ft and form the function It(x) = (fj(x) - kit) (fi(x) - k2i)... (fi(x) -knfi) and,v(x) =,6 (x),2(x)... 1r(x). Then 0t(x) = 0. The form of Vt'(x) shows that 0 =,6(x) = k + g(x) where.k $ 0 and g(x) e.3. Hence e3 contains a constant function $ 0 and e3 = 2a contrary to asumption. This lemma shows that e9 < 9a for some a. If e is maximal e = t3a. Thus the correspondence a --> 23a is (1-1) between S' and the space M of maximal two-sided ideals in 2t. Let F' be a closed subset of S' and let F be the corresponding set in M. The intersection SF = IN3 for all a e F' is the set of functions f such that f(a) = 0 for all a e F. Let 5.3b be a maximal two-sided ideal containing Zps. If b e F' there is a function 4 such that +(a) = 0 for all a in F' but +(b) $ 0. Then 4)e ZF but e 53 contrary to 5b > SF. Hence b e F' and 5% e F. Thus F is closed. Conversely let F be any closed set in M and let F' be the corresponding set in S'. It is easy to see that F' is the set of points y such thatf(y) = 0 for allf in Zps. Thus F' is the intersection of closed sets and is therefore closed. Hence the correspondence a -d 53a is a homeomorphism. THEOREM 3. Let S' be a totally disconnected bicompact space, ' a division ring and 21 a ring of g'-valued decomposition functions such that (1) 21 contains the constants and (2) if a $ b in S', then 21 contains a function f such that f(a) $ f(b). Then the space M of maximal two-sided ideals of 21 is homeomorphic to S'. If 2' is commutative, M = S. It is an open question whether or not this is true in general. Let However, it is clear that M = S since the intersection of the set of maximal two-sided ideals is (0). Theorem 3 shows that two rings 21 and 212 of the type considered are not isomorphic unless the spaces S,' and S2' are homeomorphic.' Since the division ring ft' is isomorphic to for any maximal two-sided ideal

6 338 MATHEMATICS: KASNER AND DE CICCO PROC. N. A. S. Zi in gt, the isomorphism of 91, and 512 implies that of S1' and 92'. If!R = Si the complete ring of finite decomposition functions then the converse holds. Hence necessary and sufficient conditions that Si and e2 be isomorphic are that SI' and S2' be homeomorphic and that Rl' and R2' be isomorphic. I "The Radical and Semi-simplicity for Arbitrary Rings," Amer. Jour. Math., 67, (1945). The results that we state without proof in this section can be found in the above-mentioned paper. 2 We use the notations V and A, respectively, for the logical sum and logical product of a finite number of sets. 3 Loc. cit. in reference 1, p "Applications of the Theory of Boolean Rings to General Topology," Trans. Amer. Math. Soc., 41, (1937). 5 "'Ober verschiedene Methoden der Einfuhring der Topologie in die Menge der maximalen ideale eines normierten Ringes," Math. Sbornik, 51, (1941). 6 Loc. cit. in reference 1, p Loc. cit. in reference 5, pp Loc. cit. in reference 4, p It is assumed here that the field of residues mod 2 is endowed with a T1-topology. This means that each point is both open and closed. CONVERSE OF PTOLEMY'S THEOREM ON STEREOGRAPHIC PROJECTION By EDWARD KASNER AND JOHN DE CiCco DEPARTMENTS OF MATHEMATICS, COLUMBIA UNIVERSITY AND ILLINOIS INSTITUTE OF TECHNOLOGY Communicated August 22, The Problem.-The famous theorem of Ptolemy on stereographic projection states that the perspectivity of a sphere from a point of the sphere upon a plane perpendicular to the diameter of the sphere determined by the given point is conformal. The question naturally arises as to the existence of any other surfaces for which there exists a perspectivity upon a plane which is conformal. We prove that there does not exist any other surface with this property (except for the obvious case of a parallel plane). Our result may be stated as follows: THEOREM. If a perspectivity of a surfacefrom afixed point to a fixed plane is conformal, then the surface must be a sphere; furthermore, the sphere must pass through the fixed point and have its center on the perpendicular from the fixed point to thefixed plane. Thus the only perspective conformalities upon a plane are Ptolemy's stereographic projection of the sphere (or the limiting case of the plane). 2. Beginning of the Proof of the Theorem.-Let (x, y, z) denote cartesian