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3- (21 - Y) - ( -(), E Is Primitive in 2 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, 357-476 (1939). See particularly Appendix II, Localness of Measles, 469-476. A TOPOLOGY FOR THE SET OF PRIMITIVE IDEALS IN AN ARBITRARY RING By N. JACOBSON DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY Communicated July 11, 1945 1. 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 two- sided 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 par- ticular 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 non- commutative ring there may exist primitive ideals that are not maximal. For example, the ring 2 of all linear transformations in an infinite dimen- sional 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. Downloaded by guest on September 28, 2021 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 primi- tive. 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 de- fine 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 eand3 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 com- mutative 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: Downloaded by guest on September 28, 2021 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 21 - 21 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 21 - 21. Since this correspondence preserves intersection it is a homeo- morphism between S1 and T. Suppose in particular that 2h = T the radical of 21. Then every primi- tive 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 con- sider 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 primi- tive 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 2A1.A 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 con- tinuous 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. Sup- pose 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 FaI is a set of closed sets in S, HFaF. O ifand 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,, w3if 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.
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