ON THREE PROBLEMS CONCERNING NIL-RINGS

Transcription

1 ON THREE PROBLEMS CONCERNING NIL-RINGS JAKOB LEVITZKI 1. Itroductio. I the preset ote three problems cocerig il-rigs are proposed ad certai relatios likig these problems to oe aother are discussed. First problem. The sum of all two-sided 1 il-ideals of a rig S has bee defied by G. Koethe [2, 3] 2 as the radical of 5, provided that this sum cotais also all oe-sided il-ideals of S. We shall heceforth refer to this radical as the it-radical 3 of S. It is a ope questio whether or ot there are rigs i which the 2 -radical does ot exist. Secod problem. A rig T is called semi-ilpotet (see [3, 2]) if each fiite set of elemets i T geerates a ilpotet rig. A rig which is ot semi-ilpotet is called semi-regular. Each semi-ilpotet rig is evidetly a il-rig. It is a ope questio whether or ot there exist semi-regular il-rigs. As may easily be see, this problem is equivalet to the questio whether or ot there exist semi-regular il-rigs which are geerated by a fiite set of elemets. Third problem. A il-ideal P of a rig S has bee termed by R. Baer [l, l] a radical ideal if the quotiet-rig S/P does ot cotai ilpotet ideals other tha zero. The sum U(S) ad the crosscut L(S) of all radical ideals of a rig 5 are agai radical ideals which are called the upper radical ad the lower radical respectively (see Baer [l, 1 ]). As idicated by Baer ideals may exist betwee U(S) ad L(S) which are ot radical ideals. R. Baer has also costructed a iterestig example which illustrates this possibility. Our results i the preset ote show that this pheomeo ca ot be cosidered as a exceptio to the rule but o the cotrary rather as the rule itself, ad thus the followig problem presets itself : Are there or are there ot rigs S i which U(S)'Z>L(S) ad i which furthermore each ideal which lies betwee U(S) ad L(S) is also a radical ideal? I the preset ote the followig results are obtaied: Suppose that S is a rig i which the ic-radical does ot exist, the S cotais a ifiite umber of right ideals as well as of left ideals which are semi-regular il-rigs (see Theorem 4 i 3). Suppose that S is a rig with a semi-regular upper radical Z7(5), the 5 cotais a subrig S' so that E/(S)3S" = U(S')DL(S') ad so that S' cotais a ifiite Received by the editors April 3, We shall write heceforth i short ideals istead of two-sided ideals. 2 Numbers i brackets refer to the Bibliography at the ed of the paper. 3 For the sake of coveiece we shall reserve i this paper the term radical for the sum of all semi-ilpotet ideals of the rig (see [3, 2]). 913

2 914 JAKOB LEVITZKI [December umber of ideals betwee U(S') ad L(S') which are ot radical ideals (see Theorem 5 i 3). If S is a rig with a semi-ilpotet upper radical U(S), ad if U(S)Z)L(S) 1 the 5 cotais a ifiite umber of ideals betwee U(S) ad L(S) which are ot radical ideals (see Theorem 6 i 3). Remarks. From Theorem 4 it follows that if the first problem is aswered i the affirmative the this will apply also to the secod problem. From Theorem 5 it follows that if the secod problem is aswered i the egative the this will also apply to the third problem. The above-metioed example of R. Baer is of a il-rig U of which it ca be easily proved that it is semi-ilpotet. I fact, the proof for the semi-ilpotecy of U is implicitly cotaied i Baer's proof for the illity of U (see [l, 2]). Thus our Theorem 6 explais the pheomeo which was described by Baer i his example. This theorem i cojuctio with Theorem 5 seems to justify the cojecture that the aswer to the third problem is i the egative. Notatios. If the rig T is geerated by the fiite set of elemets ht hy, t i the we write 7"= {h, h,, t } or T= {, ti, - }. The sum 4 of a fiite umber of right ideals (left ideals) Au Ai, -, A will be deoted by Ai+A%+ +A X*-i^*«The ideal, the right ideal ad the left ideal i the rig 5 which are geerated by a fiite set of elemets au a%,, a will be deoted by (au #2,, a ), (au a 2,, a ) r ad (a u a 2,, a ) t respectively. If A = (au Ö2,, a )r, the A 3 23?.ia<5. I case A = XX î 0 *^ we say that A possesses a proper right basis. Similarly a proper left basis is defied. If A = (au Ö2, >a )>the A^^y^S-aiS. I c&se A =^? am isais we say that A possesses the proper basis au a$,, a. If the rig 5 has a idetity elemet, the all bases are of course proper bases. 2. O il-ideals with proper bases. I this sectio we shall derive certai properties of il-ideals with proper bases which will be used i 3 for the proof of the assertios made i the itroductio. THEOREM 1. If Ris a ozero il-ideal with a proper right basis i a rig S, the R 2 CR- PROOF. By assumptio we may put R= X)?-! 0»»^» where a^r, i = l,,. Sice further SaiSQR, we have ^ = 1 ^. 1 ^ 5 = XXi a *'^ =^2> a d hece i? 2 = X)?-i a»^«now suppose that R 2 ( R, the by R 2 QR we would have R 2 = R, that is, We shall ot use direct sums i this paper.

3 i 9 45l ON THREE PROBLEMS CONCERNING NIL-RINGS 915 (i) R «E «A «< e *, * - i,.». I case w = lwe obtai R = air, which by successive left multiplicatio yields R = a\r for each k y that is, aj^o for each k> which is a cotradictio to the illity of JR. NOW suppose that ^ 2 ad defie w so that w m 1 (2) R = X) «A * 3 Z **R, 2 ^ m g». By puttig T ^fz\air we the have (3) JK = T + a m R, a m 5* 0. By (3) it follows that a m ca be represeted i the form (4) a m = h + a m c, h G T, c G R. Now suppose that for a certai r it has bee proved that a m = & r +# m c r, brçît, the by (4) this would yield a w = & r +(6i+a m c)c r = 6 r+1 +ömc r+1, where b r +i = b r -+-bic r. I view of the fact that T is a right ideal, it follows that &r+ie2"\ ad thus by iductio we have proved that for each 5 the elemet a m has a represetatio of the form (5) a m = b 8 + a m c\ b t G T, c G R. Now take for 5 the idex of the ilpotet elemet c, the c* = 0, ad we have a m = b s Ç:T y ad hece a m RQT t or R= ^."/a^, which is a cotradictio to (2). Cosequetly equatio (1) is impossible, which implies that R 2 (ZR- THEOREM 2. If Ris a ozero semi-ilpotet right ideal with a proper right basis i a rig S, the R 2 QR. PROOF. For a certai iteger we have by assumptio R = X)?-ia t -5 with a^r, i = l 1 - -,. Now deote by N the radical 3 of»s, the RQN, ad Sa { SQN (see [3, 2]). Hece we have R 2 = J^iaiSa k S Q^^iaiN. Now suppose that R 2 ( R, the we would have R 2 = R, that is, RQ JX^diNQR, or (6) R = 2 ^> a< E.R, i =* 1,, w. From (6) follows

4 916 JAKOB LEVITZKI [December (7) ai = ]T) a k b ik, b ik G N, i, k = 1,,». fc~l By cosiderig the rig B = {, &a, }, we have i view of (7) the relatio^ G XX.ia*B,i==l,,», which by right multiplicatio yields a{bq ^Cï-i 0^2» ^ = 1, * *, w, ad hece a*g 2*-i a *-^2» i = l,,». Thus by successive right multiplicatio we obtai for each m (8) öig^ a k B m, i = 1,,», fc=»i which implies that B m 9 0 for each m. But this is a cotradictio, sice N is semi-ilpotet, BÇ2N, ad J3 is geerated by a fiite set of elemets. Remark. Each il-ideal of a rig S is cotaied i the upper radical U(S) of 5. Oly a slight modificatio of the proof of Theorem 1 is ecessary i order to exted the validity of that theorem to oe-sided ideals of S which are cotaied i U(S). I this geeralized form, Theorem 1 would iclude Theorem 2 as a special case, sice each semi-ilpotet oe-sided ideal lies i the radical 3 N of 5, which i tur is a subset of the upper radical. THEOREM 3. If Ris a ozero semi-ilpotet ideal with a proper basis i S, the R 2 CR. PROOF. For a certai iteger we have by assumptio R~ XXiSatS, aiçir, i~l,,». If, agai, N deotes the radical 3 of S, the SaiSQN for each i, ad hece we have SaiSSa k SSa 3 -S QNajcN for ay triple of idices i, j, k. Thus it follows that # 3 = JOUj-iSaiSSatSSajSQ J^^NauN. Now suppose that R 2 <tr f the we would have R 2 = R, ad hece also i? 3 = jr, which by R = R Z Q Jjt.iNa h NQR implies (9) R = Na k N, a k G R, k «1,,». &-i By (9) it follows that each a k has the form (10) a k = X) baaie, b ki G iv, CK G iv, *, * = 1,,». Puttig JB = {, &a, } ad C= {, Ca, } we ca write (10) i the form a k Çz X^-i-^0^» fe = l,,», which by successive right ad left multiplicatio yields

5 19451 ON THREE PROBLEMS CONCERNING NIL-RINGS 917 (11) a k G Ê B"o», k - 1,,». *' 1 By (11) it follows that B m^0, O^O for each m, which is a cotradictio to the semi-ilpotecy of N. 3. Proof of statemets made i the itroductio. The proof of the followig Theorem 4 follows directly from the author's results i [3, 2], while Theorems 5 ad 6 are based o our results i 2. THEOREM 4. If S is a rig i which the K-radical does ot exist, the S cotais a ifiite umber of right ideals as well as of left ideals which are semi-regular il-rigs. PROOF. First ote that 5 cotais right il-ideals as well as left il-ideals which are ot i the upper radical U(S) of S. Ideed, by assumptio S cotais a oe-sided il-ideal A so that A < U(S). Now defie the elemet a so that aç^a but a( U(S), the the right ideal {a) r ad the left ideal (a)i are il-ideals which are ot i U(S). Now deote by R ay right il-ideal so that R( U(S), the also R 2 ( U(S), sice U(S) is a radical ideal. Cosequetly R cotais a elemet #i so that ai2?(t U(S). By a±rqr ad by the illity of R follows easily RZ)aiR. Now put R! = air; the i view of 2?i( I U(S) we may repeat with Ri the same procedure, ad thus by iductio obtai a ifiite sequece of right il-ideals Ri, R2, Rz,, each satisfyig the relatio i?i( U(S). Now each semi-ilpotet right ideal of S is cotaied i the radical 3 N of 5 (see [3, 2]) which is a subset of U(S). This implies that the right ideals of the sequece Ri, R2, are semiregular; sice a similar result holds for left ideals, the proof of our theorem is thus completed. LEMMA 1. If T is a rig with a fiite set of geerators, the for each positive iteger also T is a rig with a fiite umber of geerators. PROOF. Write T= { #i, #2,, am}, ad put bi lt i %t.., t i +h = ai 1 a i2 - a i+k, where 0^k< ad l'èijsm for j = l,, +k; the, as may easily be see, T = {, bi lt i t,..-,i +k, }, q.e.d. LEMMA 2. If T is a semi-regular il-rig with a fiite set of geerators, the T Z)T +l for each positive iteger. PROOF. By Lemma 1 we may put T \bi, 62, -, b r }. Now suppose that T = T +1, the by successive multiplicatio follows T ~T +m for each m, ad hece by puttig T = W we have W= W* for each k. I view of W= {bi,, b r } it follows easily that

6 918 JAKOB LEVITZKI [December W 2 = ^2 r i^ibiw=w f which is a cotradictio to Theorem 1. LEMMA 3. If P ad Q are il-ideals of a rig S ad if P2Ö, the P is a radical ideal if ad oly if P/Q is a radical ideal i the rig S/Q. I particular y Q is a radical ideal if ad oly if the zero ideal is a radical ideal i S/Q. PROOF. Deote by A a two-sided ideal of 5 so that A 2P, the by the so-called "secod law of isomorphisms" (see [4, p. 149]) we have (A/Q)/(P/Q)^A/P. Our lemma follows from the fact that A/P is ilpotet if ad oly if (A/Q)/(P/Q) is ilpotet. LEMMA 4. If the lower radical of a rig S is zero ad if the upper radical U(S) is semi-regular, the S cotais a subrig T so that U(S)'0>T= U(T)~Z)L(T) ad T cotais a ifiite umber of ideals betwee U(T) ad L(T) which are ot radical ideals. PROOF. By assumptio U(S) cotais a fiite set of elemets 0i, 02,, CL so that the rig T= {au 02,, 0*»} is a semi-regular il-rig. I view of the semi-regularity of U(T) ad the semiilpotecy of the lower radical 5 it follows that T~ U(T)'2)L{T) 1 ad that also the quotiet-rig W=T/L(T) is a semi-regular il-rig. If ow bi deotes the image of 0» i the homomorphism T~W, we may evidetly write W= {61,62,,b }. By Lemma 2 wehavew m+r CW m for ay pair of positive itegers ra, r, that is, the ozero ideal W m /W m+r of the rig T/W m+r is ilpotet. Cosequetly, W m + r is ot a radical ideal of W. Now defie for each positive iteger r the ideal A r of T by the relatios A r^l(t), A r /L(T)^W r t the A r is uiquely determied, ad i view of Lemma 3 oe of the ideals of the ifiite sequece A*, A$, is a radical ideal, q.e.d. LEMMA 5. If the lower radical of a rig S is zero ad if A is a ozero semi-ilpotet ideal of S, the S cotais a ifiite umber of ideals which are ot radical ideals ad which are subsets of A. PROOF. Deote by 01, 02,, a a arbitrary fiite set of ozero elemets of A ad cosider the ozero ideal A\ = (01, 02,, a ) of 5 which lies i A. As may easily be verified, we the have A\= ^i^iaiüiai. Now suppose that A\=Ai f the we would have Ai= XXi^i a^i Sî-i^SCil!, that is, Ax= 2j«i5a<5, which is a cotradictio to Theorem 3 i 2. Hece we have A\C.Ai, which implies that the ideal A\/A\ of the rig S/A\ is ilpotet, that is, 5 The lower radical is semi-ilpotet sice it is a subset of the radical N (see footote 2).

7 I945I N THREE PROBLEMS CONCERNING NIL-RINGS 919 Bi~A\ is ot a radical ideal. By L(S) = 0 it follows that J3O0. Sice OQBiQA, we may repeat with B\ the same procedure, ad thus (by iductio) obtai a ifiite sequece of semi-ilpotet ideals BiZ)B{DBzZ^ ' ' oe of which is a radical ideal, q.e.d. THEOREM 5. If S is a rig with a semi-regular upper radical U(S), the S cotais a subrig S f so that U(S)^DS'= UiS^'DLiS') ad so that S' cotais a ifiite umber of ideals betwee U(S') ad L(S') which are ot radical ideals. PROOF. By Lemma 3 (see also Baer [l, p. 539]) the lower radical of S* = S/L(S) is zero, while the upper radical U(S)/L(S) = U(S*) of S* i virtue of the semi-regularity of U(S) ad of the semi-ilpotecy 6 of L(S) is also semi-regular. Hece by Lemma 4 the rig S* cotais a subrig T so that U(S*)^>T*= U(T)DL(T) ad so that S* cotais a ifiite umber of ideals Af, A$, betwee U(T) ad L(T) which are ot radical ideals. Now defie a subrig S' of 5 so that S'0>L(S) ad S'/L(S) T. By Lemma 3 it follows the that 5' has the required properties. THEOREM 6. If S is a rig with a semi-ilpotet upper radical U(S), ad if U(S)Z)L(S), the S cotais a ifiite umber of ideals betwee U(S) ad L(S) which are ot radical ideals. PROOF. Sice by Lemma 3 the lower radical of the rig S* = S/L(S) is zero, it follows by Lemma 5 that the ozero semi-ilpotet ideal U(S)/L(S) of S* cotais a ifiite umber of ideals which are ot radical ideals. Our theorem follows ow as a cosequece of Lemma 3. BIBLIOGRAPHY 1. R. Baer, Radical ideals, Amer. J. Math. vol. 65 (1943) pp G. Koethe, Die Struhtur der Rige dere Restklasserig ach dem Radical vollstaedig reduzibel ist, Math. Zeit. vol. 32 (1930) pp J. Levitzki, O the radical of a geeral rig, Bull. Amer. Math. Soc. vol. 49 (1943) pp B. L. va der Waerde, Modere Algebra, vol. 1, Berli, HEBREW UNIVERSITY