BULL. AUSTRAL. MATH. SOC. VOL. 2 (1979), 259-271. I6A2I The lower radical construction or non-associative rings: Examples and counterexamples BJ. Gardner Some suicient conditions are presented or the lower radical construction in a variety o algebras to terminate at the step corresponding to the irst ininite ordinal. An example is also presented, in a variety satisying some non-trivial identities, o a lower radical construction terminating in our steps. Introduction Two o the most important questions in radical theory -are (1) Are semi-simple classes hereditary? and (2) How many steps are needed in the (Kurosh) lower radical construction? Airmative answers to (l) have been obtained or associative rings [7], alternative rings [7], and groups [6], while in these settings it has been shown that the lower radical construction stops at the wth step, where w is the irst ininite ordinal [.131, [5], [77]. On the other hand, or arbitrary rings, semi-simple classes are rarely hereditary (see [3] or an indication o just how rarely), while the lower radical construction need never stop [9]. Having hereditary semi-simple classes and never requiring more than to Received 13 March 1979-259
26 B.J. Gardner steps in the lower radical construction are "nice" properties or a universal class. The meagre evidence set out above might lead one to conjecture that these two properties go together. will be adduced in 2. Some contrary evidence In recent years there have been a number o investigations o hereditary semi-simple classes with various varieties o rings or algebras used as universal classes; see, or example, [ ], [72], [3], [2]. In the last o these, two conditions on a variety o algebras over a ield (conditions satisied by the class o associative rings) were studied, which jointly imply that semi-simple classes are hereditary. In 1 o this paper we look at the termination o the lower radical construction rom a similar point o view. In 2 we present an example o a lower radical construction which terminates at the ourth step. The universal class is the class o (nonassociative) rings R or which there is a inite series J -a J < R with associative actors, and the class on which the construction is carried out is the class o all ields (necessarily associative). Throughout this paper we shall deal with algebras (not necessarily associative) over a commutative, associative ring Q. with identity, sometimes specializing to the case where J2 is the ring Z o integers. The symbol ** indicates an ideal. A seroalgebra is one satisying the identity xy =. A universal class o ^-algebras (a setting or radical theory) is a hereditary, homomorphically closed class; in this paper, all universal classes will be varieties. We inally recall the Kurosh Lower Radical Construction (see [6] and C'3]). L e -t M be a non-empty homomorphically closed class o i-algebras (in some universal class W ). Let M. = M ; Mg = {-4 every non-zero homomorphic image o A has a non-zero ideal in M or some a < $}, where g is an ordinal greater than 1. Then L(M), the lower radical class generated by M, is U M.. 6 "
Non-associative rings 261 1. Some universal classes with u-step lower radical constructions In this section we shall consider some examples o lower radical constructions which terminate in a) steps. Our universal classes will be varieties W o i-algebras satisying various combinations o properties (Vl)-(VU) listed below. For any algebra A, we deine 9 9 " * 5 "» * " 9 and A y ' = \ A y ' i is a limit ordinal. A is solvable i a<3-4 = or some inite n. A, as usual, is the linear span o all products o length n o elements o A. I J <a J < A, we denote by X* the ideal o A generated by X. The conditions with which we shall be concerned are the ollowing: (VI) i I <i J o A Z W, then (I*/- = or some ordinal a ; (V2) i X < «7 < <4 W, then every non-zero homomorphic image o I*/I has a non-zero nilpotent ideal; (V3) i I <J <ia W, then {I*/I)^ = or some inite (VU) i X < A W, then r <i A or some integer s > 1. Varieties satisying (V3) (respectively (VU)) are called Andrunakievich varieties (respectively s-varieties). In [2] it was shown that the n and s in (V3), (VU) can be chosen independently o the algebra A. Zwier [74] initiated the study o 8-varieties; they, include the varieties o associative algebras (a 2-variety) alternative algebras (a 2-variety) and Jordan algebras (a 3-variety). In [2] characterizations o semi-simple classes were obtained or W satisying (V3) and (Vh) when i is a ield. Less complete results were obtained or more general i. This pattern repeats itsel here: lie is much simpler with algebras over a ield because there is "essentially one" zeroalgebra.
262 B.J. Gard ner The ollowing result appears in [7]; we include a proo or completeness. LEMMA 1.1. Let M be a homomorphically closed class. I I <i A, I M,, and A/I M,, then A M' 3 where t = max{c+l, 1}. Proo. Let A/K t. I J < #, then we have # J/I n X S (I+K)/K < A/K with I/I n K M k, while i I <=_ K, then X/K s (A/I)/(K/I) M z. Thus.4/i certainly has a non-zero ideal in some M with r < t, so that A M t. // The next two propositions provide most o the inormation needed or the principal results o this section. PROPOSITION 1.2. Let W be a variety o Q-algebras satisying (V2) and (Vh). Let M be a non-void homomorphically closed subclass o W containing all zeroalgebras in W. I I o J < A W and 1 (, then I* M n. s+1 Proo. J K, then Consider a non-zero homomorphic image I*/K o I*. I J/J n K (I+K)/K < I*/K, and J/J n X M = M. I Jct, then J^/X, as a homomorphic image o J*/J, has a non-zero nilpotent ideal X (by (V2)). By (VU), we can deine a sequence X/Q)> %/,\, x /o)' o idea l s o I*/K by X () = X ' X (l) = *> X {2) = ^1)' * Since X is nilpotent, there is an m such that X, < = ^ X.,.. Let X, -v = L/K. Then t L/K <ii*/k and (L/K) =. (m-1), I s (L/K) c (L/K) = and we have a series \Li/A) ^ [L/K) <]... <] \L/K) <] \.L/K)
Non-associative rings 263 with zeroalgebra actors. By Lemma 1.2, {S - X), (L/K) {S ~ 2) so (L/X) (S " 2) M 2. Similarly (L/K) {s ~ 3) M-,..., L/K = (L/K) {S ~ S) U. We have now shown that every non-zero homomorphic image o J* has a nonzero ideal in M, or some k < s + 1 ; that is that J* M. // When S] is a ield we get a stronger statement. COROLLARY 1.3. Let W be a variety o algebras over a ield satisying (V2) and (VU). Let M >e a non-void homomorphically closed subclass o W containing the one-dimensional zeroalgebra. I I < J <\A e W and IH, then J" M S Proo. Let B be a zeroalgebra. Then every non-zero homomorphic image o B has an ideal which is a one-dimensional zeroalgebra. Thus B M, so Mp ' satisies the hypotheses o Proposition 1.3. I now JED and K J o A, then J M o, so I* (. (M o ). n = M _. // PROPOSITION 1.4. Let ill be a variety o ^algebras satisying (Vl) and (vlt)j M a homomorphically closed subclass o W containing no zeroalgebras (not equal to ). I I < J <.4 W and I M 3 then I <A. Proo. 2 Since I/I is a zeroalgebra in M, I must be idempotent. The proo now runs like that o Proposition 3.5 in [2]. // Combining Corollary 1.3 with the ield case o Proposition l.u, we get PROPOSITION 1.5. Let i be a ield, W a variety o ^-algebras satisying (Vl), (V2), and (vu). I I o J < A i 111 and KM, then To see this we need only observe that a homomorphically closed class o zeroalgebras in this case either contains a one-dimensional zeroalgebra or contains no zeroalgebras. //
264 B.J. Gardner Let A be a non-zero solvable algebra in a variety W satisying (V3) and (Vh). Then A s <=_ A 2, that is, in the notation used above, A. * c A" 1 '. By induction, A,, c A" n ' or each n. But A^n' = eventually, so, or some n, we have * A (n-l) < > A i y«-l)) S =A {n) = Thus A has a non-zero nilpotent ideal. It ollows that W satisies (V2) (and, o course, (Vl)). preceding results, we get Combining these observations with the PROPOSITION 1.6. Let W be a variety o Q-algebras satisying (V3) and (vu). Let M be a homomorphically closed subclass o W containing all zeroalgebras or none. I I o J < A (. W and I M, then COROLLARY 1.7. Let W be a variety o algebras over a ield, satisying (V3) and (vu). I M is a homomorphically closed subclass o ill, J < J<3 A and JEM, then I* M ^.. // let For a non-void homomorphically closed subclass M o a variety W, = {A W every non-zero homomorphic image o A has a non-zero Then Y(M) is a radical class [7] and hence i(m) c Y(M). We can now prove our termination theorem. accessible subalgebra in M}. THEOREM 1.8. Let W be a variety o Q-algebras, M a non^void homomorphically closed subclass o W. Then L(M) = M i one o the ollowing sets o conditions holds: (i) W satisies (V2) and (v^) and M contains all zeroalgebras; (ii) W satisies (Vl) and (V^) and M contains no zeroalgebras; (Hi) W satisies (V3) and (vl*) and M contains all zeroalgebras or none; (iv) (it satisies (V2) and (Vk), U contains a zeroalgebra, and (i is a ield;
Non-associative rings 265 (v) bl satisies (Vl), (V2), and (Vk), and Si is a ield; (vi) W satisies (V3) and {vh), and Si is a ield. In (ii) and in (iii), (iv), and (v), where M consists o idempotent algebras, we have (M) =Mp. Proo. Reerence to the earlier results o this section establishes that in (i)-(vi), i I <i J o A t W and I E (I, then I* M g+2. I i? (M), then i 7(M) ; so or every non-zero R/L we have a inite series # M < A/ <... < M < M = #/L with M M. Consider M <i M <t M. We have M* t M o. Arguing n n n ± YL C. n o+ by induction, we show that i?/l has a non-zero ideal in M, or some, inite k. Thus R t M. to This last assertion o the theorem ollows rom Proposition l.u. // As noted above, the varieties o associative, alternative, and Jordan algebras satisy (Vh). By Andrunakievich's Lemma ((J*/J) = o), the associative algebras satisy (Vl), (V2), (V3). Results in [4] show that the alternative algebras satisy (Vl) and (V2). I % Si, then the Jordan algebras satisy (V2). To see this we use the ollowing result. THEOREM 1.9 (SI in'ko [72], Theorem l). // A is a Jordan algebra over a ring Si containing h, i I < J < A, and i J/I has no nilpotent ideals, then I < A. PROPOSITION containing h satisies (V2). 1.1. The variety o Jordan algebras over a ring Si Proo. Let I <x J <3 A. Any homomorphic image o I*/I is isomorphic to I*/K or some K <c I* with J c K. I I*/K has no nilpotent ideals, then by Theorem 1.9, K < A. But then (since IcK ) I* = K. II Varieties satisying (V3) and (vu) are discussed in [2]. The main drawback, o course, in the approach we have used in this section, is that "complete" results can be stated only or algebras over a
266 B. J. Ga rd ner ield (see [2]). I some way can be ound to take account o the additive structure o algebras, the method may produce new and simpler proos o known results (or example, or alternative rings) as well as new inormation. 2. A 4-step termination It is not known whether, in the universal class o associative rings, there exists a class M with I(M) = M ^ M or 3 < k < u ; in act no example appears to have been given in any universal class o a class with this property. We present such an example here. Let A denote the variety o associative rings (2-algebras). Recall that or varieties l/,w, 1/oW is the variety o algebras which are extensions o (/-algebras by W-algebras. THEOREM 2.1. Let F be the class o (associative) ields. Then in (A A)oAjWe have Proo. with basis Let K be a inite prime ield and let A be the X-algebra {e, /, g} and multiplication table e / g e e / / a e 9 Let ± I < A. I oe + b + eg I, then Hence ae + eg I, so b = (b+ce) = {(ae+b+cg)) J. eg = g(ae+cg) I, and thus ae (. I. I c ±, then (since K is a inite prime ield) g 6 I. But then e = g I and / = eg J, so I = A. Thus, i I t A, we have I c_<e, >, the subspace spanned by e and /. I ae + b i", then as above, ae and b (. I. I a *, then e J,so = eg (. I. Similarly, e 6 J i b t. Since J # {}
Non-associative rings 267 we have e, Z I and so I = <e, />. Now <e, >^K@K and 4/<e, l^j,so A U o A. Let A ' be an isomorphic copy o A, with basis {e 1, /', } and multiplication table /' We put A and A ' together in an algebra B with basis \e,, g,, ',, hi and multiplication table e e e h g e g g g g e /' Let I be a non-zero ideal o B and let /' g h h a = ae + b + eg + a' + b'' + a' + dh be in I. Multiplying on the right by, we see that I contains ae + b + eg + a' + b'' + + dh = a' + a' + dg. Then multiplying on the let by, we get (1) a' + ey<?' + dv = a' + Cy I. Hence a' = (a'+c') (. I, so by (l), (2) cy e i. Multiplying a on the let by, we get b' + cv = ae + V/ + eg + a'gr' + 2>'? ' + c' + d/z I.
268 B.J. Gard ner Then (2) implies that b 'e ' I, so b ' = b 'e ' J. We have thus ar shown that a ', b ', a ' I. A similar argument shows that ae, b, eg t I, whence also dh d I. I d #, then h J, and then (as can be seen rom the table, e = he ' and so on), e,, g,, /', I. Thus i I t B, then 1 e ( e,, 9, e>,, ' } = A A'. Since I t, the argument used above shows that J contains at least one o e,, g,,,. I e I, then = eg i I, g = he I, = eh Z I, = I, = h Z I, so I = A @ A'. I / J, then e=/ J,so J = 4 /4'. I g I, then e = j / I, so again J = A A'. Similarly I = A A' i,/',or I. The ideals o B are thus, A A', and B. As noted beore, the ideals o A are, J = (e, ), and A, so the ideals o ^4' are, J' = <, /'>, and A'. Since B/A @ A' ^ K, we have B (A o A) o A. Now J, J' = K F 2\F Since A/c7 = F we have,4 F by Lemma 1.1. Similarly A' F. In act, A and /I' F^Fo ' since they have no simple ideals. j t Consider A A '. We have J, j' < A A, then. Let A A'/L #. I * J/(J n ) 3* (J+L)/L< A 4'/L, and c7/(</ n L) F 2. Similarly /I A'/L has a non-zero ideal in F 2 i J' ^ L. I contains both J and j', then 4 A'/L is a homomorphic image o U A' )/(j j') ^ K ^. Thus A A' F 3. Since B/A A' = K F, Lemma 1.1 says that S F,. Now A A' \ F (since A, A 1 F J, and 5 F (since its non-zero ideals are B and A A', and neither is simple). Thus B has no non-zero ideals in F, whence B $ F and thus S F, \F. We complete the proo by showing that L(F) = F,. Let R be in L(F). Then R has an ideal L such that L A o A and R/L A, and thus L has an ideal M such that W A and L/M A. Now F is
Non-associative rings 269 hereditary, so L, M L{ F), and also L/M and R/L L( F). Now every non-zero homomorphic image o M has a non-zero ideal in F, and since M is associative, this means that M F. In the same way I/W F 2, so by Lemma 1.1, L F Since also R/L F, a urther application o Lemma 1.1 shows that R (. F,. // We note that there is some similarity between our construction and that used by Ryabukhin [9] to show that the lower radical construction in the class o all rings need not terminate at all. Our ring B diers, however, rom the ourth ring in Ryabukhin's transinite sequence o rings. The argument leading up to Theorem 1.8 closely parallels those used in the proo that L(M) = M or any class M o associative rings (Sulinski, Anderson and Divinsky [73]), alternative rings (Krempa [5]), and also groups (Scukin [77]). It essentially involves showing (i) that i T is an accessible subring o R, then the ideal T o R generated by T is not "radically" dierent rom T, and (ii) that then y(m) c L(M) (and thus 7(M) = L(M) ). The construction o (F) (in (A o A) o A J, which we have just discussed, does not conorm to this pattern, even though it terminates ater a inite number o steps. THEOREM 2.2. In (A o A) o A, L(F) # y(f), and L{ ) does not have a hereditary semi-simple class. Proo. It is shown in [3] (Theorem 2.1 (iii)) that a hereditary radical class R o the orm Y(C) (called in [3] a hereditary homomorphic orthogonality radical class), in A A, satisies the condition (3) i?(r«/ a where R is the zeroring on the additive group o R. Condition (3) is satisied also by (A o A) o A (the proo is quite similar). Since L(F) is hereditary and does not satisy (3), it can not coincide with 7(F). The second assertion o the theorem can be obtained rom Theorem 2.1
27 B.J. Ga rd ner (i) o C3] in the same way. // Thus - reerring back to our questions in the Introduction - a lower radical construction over a "quite small" class can terminate in a inite number o steps and yet produce a radical class with a non-hereditary semisimple class. How quickly a lower radical construction must terminate when the associated semi-simple class is hereditary remains unknown. Reerences [7] T. Anderson, N. Divinsky, and A. Sulinski, "Hereditary radicals in associative and alternative rings", Canad. J. Math. 17 (l 65), 59U-63. [2] Tim Anderson and B.J. Gardner, "Semi-simple classes in a variety satisying an Andrunakievich Lemma", Bull. Austral. Math. Soc. 18 (1978), 187-2. [3] B.J. Gardner, "Some degeneracy and pathology in non-associative radical theory", Arm. Univ. Sci. Budapest. Eotv'os Seat. Math. (to appear). [4] I.R. Hentzel and M. Slater, "On the Andrunakievich Lemma or alternative rings", J. Algebra 27 (1973), 21*3-256. [5] J. Krempa, "Lower radical properties or alternative rings", Bull. Acad. Polon. Sai. Sir. Sai. Math. Astronom. Phys. 23 (1975), [6] A.G. Kuros, "Radicals in the theory o groups", Rings, modules and radicals, 271-296 (Colloq. Math. Soc. Janos Bolyai, 6. North- Holland, Amsterdam, London, 1973). [7] W.G. Leavitt and Yu-Lee Lee, "A radical coinciding with the lower radical in associative and alternative rings", Paciic J. Math. 3 (1969), k59-^62. [S] A.A. Nikitin, "Supernilpotent radicals o and Logic 12 (1973), 173-177. (-1, l)-rings", Algebra [9] Ju.M. Ryabukhin, "Lower radical rings", Math. Notes 2 (1967), 631-633.
Non-associative rings 27 1 [7] M. Sadiq Zia and Richard Wiegandt, "On the terminating o the lower radical construction", typescript. [JI] H.H. 1%HMH [K.K. Scukin], "H TeopHM palmha/iob B rpynnax" [On the theory o radicals in groups], Sibirsk. Mat. 2. 3 (1962), 932-9l»2. [J2] A.M. SI in'ko, "On radical Jordan rings", Algebra and Logic 11 (1972), 121-126. [13] A. Sulinski, T. Anderson and N. Divinsky, "Lower radical properties or associative and alternative rings", J. London Math. Soc. 41 (1966), U17-U24. [J4] Paul J. Zwier, "Prime ideals in a large class o nonassociative rings", Trans. Amer. Math. Soc. 158 (1971), 257-271. Department o Mathematics, University o Tasmania, Hobart, Tasmania.