Math. Ann. 194, 27--34 (1971) @ by Springer-Verlag 1971

The Levitzki Radical in Varieties of Algebras*'** TIM ANDERSON

Introduction

Recently Zhevlakov [4] has shown that for Jordan rings free of 2-torsion, the property of being locally nilpotent is a radical property (in the sense of Kurosh-Amitsur [1]), while Tsai [3] has added an analogue of the Babi6 Theorem for these rings. These results on the Levitzki radical are also valid for alternative rings. Indeed, more generally, Zwier [5] has shown that if the defining identities for a variety of rings obey certain combinatorial laws, common to the Jordan and alternative cases, then there is a satisfactory theory of the Levitzki radical in the variety.

In this paper we consider the general problem: What are necessary and sufficient conditions for the existence of the Levitzki radical in a variety of rings?

We give a partial solution to this problem by formulating a condition on the universal enveloping tings defined for a variety. As in the case of Zwier's work, the theory of the Levitzki radical for alternative rings follows as a special case of a more general result. However, our approach differs from Zwier's in that we study the defining identities of a variety from an implicit point of view.

1. Preliminary Definitions and Discussion

Throughout this paper, Q will be a fixed but arbitrary commutative associative ring with unity and the term "algebra" will always mean "Q-algebra". We recall that a Q-algebra is a non-associative ring A which is also a unital left Q-module for the ring Q and such that ct(xy) = (~x)y = x(Qty) for all x, y ~ A and ~t E ~. In particular, if Q is taken to be the ring of integers, then any ring is a Q-algebra. Thus, our results, while stated for Q-algebras, will be equally valid for rings. The term "ideal" will always mean "two-sided ideal" and we shall write I < A to denote the fact that I is an ideal of A. Finally, whenever we use the expression "linear", we shall mean "~-linear".

Let A n denote the linear span of all products of n elements of an algebra A. The algebra A is said to be nilpotent if An= 0 for some n. More generally, A is called locally nilpotent if every finitely generated subalgebra of A is nilpotent.

It is convenient to describe the nilpotence of an algebra A in terms of its right and left multiplications R a and Lo respectively, where R a : x - ~ x a and

* This research was partially supported by the National Research Council of Canada. ** The author gratefully acknowledges the use of facilities so kindly extended him by the

University of Barcelona and the University of Birmingham as this paper was being written.

28 T. Anderson:

L , , : x - , a x for a, x E A. By induction it may be shown that any product of 2 r dements of A can be written in the form (x) Ya~ ... Y~, for some x, at . . . . . ar~ A, where Y denotes ambiguously R or L. Thus A is nilpotent if and only if there exists an integer r such that Y,l --" Y ,̀~ = 0 for all al . . . . . a, e A.

A weaker notion than nilpotence is that of solvability. For an algebra A we define inductively A ~") by A t°) = A and A ~") = (A ~"- 1))2 for n > 0. While A ~1) is an ideal of A, in general A t") is only a subalgebra of A. If A t") = 0 for some integer n then A is said to be solvable. It is easy to see that A 2r ____ A ~'), so that nilpotent algebras are always solvable.

A technical difficulty which will arise in our work lies in the fact that a variety may be described by several sets of identities. For this reason we need to recall some basic facts about identities.

Let X = {xt, ..., x, . . . . } be a countably infinite set and #{X} be the free non-associative algebra based on the set X. The algebra q~{X} may be charac- terized by the property that any map 0 : X - , A , where A is an algebra, can be uniquely extended to a homomorphism if: #{X} - ,A. An algebra A is said to satisfy the identity p ifp e #{X} and for every homomorphism r/: #{X} - ,A, (p)q =0. In view of the above mapping property for #{X}, this is equivalent to saying that p(at . . . . , a , )= 0 for all a t , . . . , a, e A if p = p ( x t . . . . . x,). Given a subset I of #{X}, the variety C(I ) defined by I consists of those algebras which satisfy all the identities in 1.

An ideal of ~{X} which is invariant under all the endomorphisms of the algebra #{X} is called a T-ideal. Evidently, if I = #{X} and I ' is the T-ideal of # {X} generated by I then ~¢~'(I) = ~(I ' ) . The importance of expanding I to I' lies in the following fact: Ifp is an identity for each A e C(I ) then p e I'. To see this, it is sufficient to note that # { X } / I ' is contained in V(I).

Most studies of algebras take place within varieties defined by homogeneous identities, and in this paper it will always be assumed that the defining identities are homogeneous m all variables. Moreover, in order to make certain degree arguments later, we shall have to require that the identities in the T-ideal I' generated by I are essentially homogeneous as well. Therefore, we now make the additional assumption that the linearizations of the identities of I are contained in I'. Then the structure of ¢~{X} as a graded algebra (with respect to the degree function) carries over to the algebra # { X } / I ' , and we have the following result: If p(x l . . . . . x ,) ~ I ' and p = Y, pj where pj is homogeneous of degree j in xi then pje I ' for allj.

I fA ~ ~ ( I ) and M is a #-space which admits bilinear compositions ma and am(EM) for a e A , rn~ M then we can form the split null extension A @ M by defining (a + m) (b + n) = ab + an + mb for a, b e A, m, n e M, If the algebra A q ~ M is contained in C( I ) then M is called a bimodule (more precisely, an I-bimodule) for A. The notion of bimodule is equivalent to that of representa- tion. That is, given a #-space M and a pair of linear maps a - , S`,, a---, T~ of A into Hom(M,M), bilinear compositions ma and am can be defined by ma = (ra)So and am = (m) T,, for m e M, a e A. If, then, M is a bimodule, the pair of maps a- ,S , , , a - , T`, is called a representation of A in Horn(M, M) (in fact,

The Levitzki Radical in Varieties of Algebras 29

we may speak of a representation of A in any associative algebra X since in any case, X is isomorphic to a subalgebra of Hom(X, X)).

For A E Y(I) we may form ll(A), the universal enveloping algebra for the representations of A. The associative algebra R(A) has the property that there is a (canonical) representation a--, S*, a--, To* of A in II(A) such that for any representation a ~ S , , a - , Ta of A in an associative algebra X there exists a unique homomorphism ~p : ~(A)---,X such that S = S*~p and T = T*~. This property will be referred to as the universal mapping property for II(A).

Another property we shall need is that II(A) is generated by the elements S*, To*, a e A. To verify this, as well as the existence and uniqueness of R(A), we refer the reader to Jacobson [2]. However, we caution the reader that our definition of ~I(A) differs slightly from Jacobson's in that our II(A) does not contain a unity element.

2. Main Results

(2.1) Lemma. Suppose the variety 3e'(I) has the following property: (J[) Every solvable B ~ ~ ( I ) is locally nilpotent. Then the free algebra A in ~ ( I ) generated by the set {y, xl . . . . . x,} has the

Jbllowing property: (2.2) For each integer k >= 1 there is an integer f (k ) such that any product

of the form (y) Y,~... Y,,k~, where {a~ . . . . . aftk) } C= {X 1 . . . . . X,} and Y = R or L, can be written as

(Y) Y,~ ... Y~,,,,, = Y' ~(Y) Y~,,, ... Yc,,,,,,,

where the ~i ~ ~, the cj, i' s are monomials in x 1 . . . . . x,, and for each i, at least one of cl.i, ..., Cm,,i is of degree at least 2 k-1 in the xfs.

Proof. Consider the algebra A/Tk, where T k is the ideal of A generated by a (k). As T k D__ A (k), A/Tk is solvable and finitely generated; hence A/Tk is nilpotent by condition (J/). Thus, there is an integer f (k ) such that (a)Ya,... Ya,k, ~ Tk for all a, al, . . . , a fiR ) ~ A, where Y= R or L, Evidently then, (a)Ya~ ..: Ya,k~ -- Z u'iv' i + Z (uivi)P i, where u' i, v'i, ui, vi are in A tk-1) and the P{s are products of Yb~, 's, where the elements bj, i are in A and Y = R or L. In this relation put a = y and choose {al . . . . . aftk) } C {X~ . . . . . X.}. Then the left hand side of the resulting relation is of degree one in y, hence so too is the right hand side. Using now the fact that A(k-I)C-_A 2k-i, w e obtain (2.2).

(2.3) Theorem. Let ~/'(I) be a variety satisfying the condition (Jg). Then U(B) is nilpotent for each finitely generated solvable algebra B in ~IP(I).

Proof 1. Since B is finitely generated and solvable, B 2 k - ~ - 0 for some k, because of (Jr'); hence there exist b~ . . . . . b, e B such that every element of B is a linear combination of b 1 . . . . , b.. The algebra II(B) is generated by the elements S*,, T{,*, where b---,S'~, b---, T* is the canonical representation of B in

The author is thankful to the referee for suggesting improvements in the following proof.

30 T. Anderson

II(B). For the bimodule M belonging to the canonical representation, the split null extension B ~ M is in ~( / ) .

From a fundamental property of free algebras [2, p. 26], the relations (2.2) are identities for all the algebras in ~r'(I), and in particular, for B @ M. Choosing, in (2.2), y E M and {xl, ..., x,} = {bl . . . . . b,}, and using the fact that B 2~-1 =0, we obtain the result that Yo, ... Yo:~k~ = 0 for {a 1 . . . . . a:~k)} ~_ {bl, . . . , b,}, where Y = S* or T*. Therefore II(B) :~k) = 0 and II(B) is nilpotent.

A variety ~ ( I ) for which there exists an integer s ~ 2 such that A ~ ~ ( I ) and J < A imply J ' < A has been called an s-variety by Zwier I-5]. Let us note that for a 2-variety ~(I ) , A ~ ~ ( I ) and J < A imply j k < A for all k ~ 2. To see this, consider the free algebra F in 4/:(1) generated by {xl, x2, x3} and let P be the ideal o f F generated by x 1 and x 2. Then x~x 2 ~p2 and (Xxx2)x3 E p2 because C(I) is a 2-variety. Then from the homogeneity conditions on I, it follows that (xl x2)x3 = f ( x l , x2, x3), where f ( x l , x2, x3) is a linear combina- tion of the terms (x3)Ix, Yx~ and (x3)Y~ Y~,, where Y = R or L. Similarly, x3(x lx2)=g(xt , x2,x3), where g(xl ,x2,x3) is composed of the same terms above. Then using these two identities and induction, it can be shown that J < A ~ ~ ( I ) implies j k < A for all k ~ 2.

We now consider the following condition for 2-varieties: (jlr) H(A) is nilpotent for each finitely generated trivial A ~ C(I).

(2.4) Lemma. Let ~v'(I) be a 2-variety satisfying (jtr). Suppose A is a finitely generated algebra in 3v'(I). Then for each integer k > 1 there exists an integer re(k) such that ((A2) k) Eat.. . Yo,,~ ~- (A2) k+ 1 for all a l , . . . , am(k) ~ A, where Y is ambiguously L or R.

Proof. The algebra A/A 2 is trivial and finitely generated; hence II(A/A 2) is nilpotent by (,4:). Because ~V(I) is a 2-variety, (A2) k and (A2) k+l are ideals of A. Moreover, (A2) k+ 1 contains (A2) k . A 2 and A 2 . (A2)~. Hence (A2)k/(A2) k+ 1 is a bimodule for the algebra A/A 2 and we may represent A/A 2 in the algebra of linear transformations of the space (A2~'/(A2) k÷ 1.

Specifically, for a e A, we let Sa2 +o : (A2) k+ 1 + x ~ (A2) k+ 1 + xR , and TA~+o : (A2) k+l + x ~ ( A 2 ) k+l + xLo, where x e (A2) k. Then the pair A 2 +a~Sa2+o, A2+a~Ta~+o is the representation of A/A 2 provided by the bimodule (A 2)k/(A 2) k+ 1.

Using the universal mapping property for II(A/A2), we conclude that there is a homomorphism ~p:I I (A/A2)~X such that S = S * w and T = T*~p where S* and T* denote the canonical representation of A/A 2 in II(A/A 2) and X is the associative algebra generated by S,t2+a, Ta~+,, a t A. Therefore, tp maps II(A/A 2) onto X and X is nilpotent. This implies there is an integer m(k) such that ((A2)~)Yal ... Y~,,~k,_~(A2~ +1 for all al, ...,am(k)~A , where Y = R or L.

The idea of the next lemma is due to Zhevlakov [4].

(2.5) l.emma. Suppose for each finitely generated algebra A ~ ~v'(I) there exists an integer m such that A s ~_ A 2 A 2. Then if B is a finitely generated algebra in q:(1), so is B ~) finitely generated for all n.

The Levitzki Radical in Varieties of Algebras 31

Proof. Evidently it is sufficient to prove B 2 is finitely generated, and without loss of generality we may assume B to be the free algebra in ~ ( I ) generated by x 1 . . . . , Xr

Let Z be the set of all monomials of degree r in the xi's, where 2 < r < m and m is an integer such that B m ~_ B 2 B 2. We claim that Z generates B 2, and this will prove the lemma since Z is a finite set.

Suppose y is a monomial of degree greater than two which is not in the subalgebra generated by Z. Necessarily, y has degree at least m, whence y e B 2 B 2. Therefore y = Y.,~tuivi, where the ~i's are in ~ and the ui's and vi's are monomials of degree at least two. Since B is a free algebra and the defining identities of I are homogeneous, degy = degu~ +degv~ for all i. However, by induction, we may assume that the u~'s and v~'s are in the subalgebra generated by Z, whence so too is y in that subalgebra. Thus the lemma is proved.

(2.6) Lemma. Let ~ ( I ) be a 2-variety satisfying (.At). Then (1) each finitely generated solvable algebra in ~v'(I) is nilpotent, and (2) if B is a finitely generated algebra in 4/'(I) so is B ~") finitely generated for all n.

Proof. Let A be a finitely generated algebra in ~e~(I). Applying Lemma 2.4 we can choose an integer re(l) such that (A2)ya, ... Ynm~,~A2A 2 for all a 1 . . . . . amtl~eA, where Y = R or L. Thus for all x, a o , a 1, ..., amtl) e A , (X) YaoYal . . . Yam~I)CA2A2, where Y = R or L. If we put m = 2 m(1)+1 then we have A"C=A2A 2. l~ow Lemma 2.5 gives the result (2).

To prove (1) we use induction on the integer t for which A t') = 0 for a solvable algebra A, the result being trivial for t = 0 or 1. Suppose A is an arbitrary finitely generated solvable algebra in "//'(I). Then from (2) A 2 is also finitely generated and solvable, and applying the induction hypothesis to A 2 we see that (A2) k÷l = 0 for some k. Put r = r e ( l )+ . . . + re(k), where the m(i)'s are chosen as in Lemma 2.4. Then for a~ . . . . . a r ~ A, we have by iteration of Lemma 2.4 that (A2) yal ... Ya~_(A2)k+l=O. Hence for all x ,a o . . . . , a reA , (x) Y,o ... Y~ = 0 and A is nilpotent.

(2.7) Theorem. Let ~e~(I) be a 2-variety. Then the following statements are equivalent:

(i) Local nilpotence is a radical property in "It(I). (ii) Every solvable B ~ q/~(I) is locally nilpotent.

(iii) U(B) is nilpotent for every finitely generated solvable B ~ "t/'(I). (iv) ~(B) is nilpotent for every finitely generated trivial B ~ q/'(I). Moreover, when local nilpotence is a radical property in ~l/'(I), every semi-

simple algebra in "//'(I) is a subdirect sum of prime semi-simple algebras in ~(1) (where "prime" is defined as for associative rings).

Proof. Even when "//(I) is not a 2-variety, if local nilpotence is a radical property in q/'(I) every solvable B e q/'(I) must be locally nilpotent. Indeed this follows easily by induction on the integer t for which B tt~ = 0 and the fact that B/B 2 and B 2 locally nilpotent imply B locally nilpotent, when local nilpotence is a radical property. Thus, (i) implies (ii), and (ii) implies (iii) because of Theo- rem 2.3. Trivially, (iii) implies (iv). On the other hand, (iv) implies Lemma 2.6,

32 T. Anderson:

and having that lemma, one can simply prove (i) as in the case of the well- known proof for associative rings. See, for example, [1] or [4].

The proof of the remaining part of the theorem, which is the Babi6 Theorem, can again be proved in exactly the same way it is proved for associative rings. See [3].

(2.8) CorolLary. Local nilpotence is a radical property in the variety of alternative algebras.

Proof. It is easy to see that alternative algebras form a 2-variety and that the defining identities have the homogeneity properties of Section 1. Thus, we only have to show they satisfy the condition (Jff).

For an alternative algebra A we have the following identities for the right and left multiplications: LaLQ=La2, RQRa=Ra~, LaLb + LbLa=Lab+ba, RaR b -1- RbR a = Rab+b a, LaR b -- RbL,~ + RaR b - Rob = O, and LaR b + Lab -- RbL a - L b L , , = O for a , b ~ A .

Suppose further A is generated by a I . . . . , a n and that A 2 =0. Then ll(A) is generated by the 2n elements Yo, where Y= S* or T* and a ~ S * , a ~ Ta* is the canonical representation of A in ll(A).

Since A 2 = 0, we have the relations (1) S~ISQ,* * -- T.*a, T,*,, = 0 for all i, and the commutivity relations (2) * * - * * * * - T* for i , j , Sa, S a j - -SajS~,, T~, T ~ j - - T~ ,, (3) T'~,Saj-SajT~,* * * * +Sa, * for all i,j, and (4) T~,* Saj*- Saj* T2,* - T~ T'a, = o for all i,j.

Consider any product P of 2n + I of the generators So*, T~*. It will be of the form e = Yal ... Yu2.÷,, where {ul, . . . , U2n+l} ~ {al . . . . . an} and Y = S* or T*. At least n + 1 of the Y's must be S* or T*. Assume, for example, at least n + 1 of the Y's are S*. Then P has the form

P . . . . s : ,

. . . * * . . . . for some j. Using the relations (2) and (3), we may assume P = SajSaj But then P = 0 because of (1). Equally, if at least n + 1 of the Y's are T*, we may show P = 0 using (2), (4), and (1). Thus II(A) 2"+' =0.

3. Open Questions

A disagreeable feature of Theorem 2.7 is the presence of the assumption that ~f(I) is a 2-variety. For example, the theorem does not apply to the important case of Jordan algebras because they form a 3-variety. However, we believe similar results should be valid for arbitrary varieties - at least for s-varieties.

In proving the existence of the Levitzki radical in ~/r(I), what has to be shown is that the notions of solvability and nilpotence coincide for the finitely generated algebras in ~(I) . In this context it should be noted that the converse of Theorem 2.3 is true. Indeed, if II(B) is nilpotent for all finitely generated solvable B e ~ ( I ) then in particular the regular representation of B is nilpotent,

The Levitzki Radical in Varieties of Algebras 33

whence B itself is nilpotent. However, in practice it would be very difficult to verify this condition on the universal enveloping algebras of a given variety.

It may be advisable to study the lower radical determined by the trivial algebras in ~ ( I ) in order to arrive at other, more suitable, conditions for the existence of the Levitzki radical in ~( I ) . That is, we put ~t'o = {A ~ ~ ( I ) IA 2 = 0} and define inductively ,/ / , to be the class of all algebras A e i//(1) with the property that each non-zero homomorphic image of A contains a non-zero ideal in J / , _ 1 for 0 < n < ~ . It is easy to see that if A ~") = 0 then A e J t , . Thus if it can be shown that the algebras in Jr', are locally nilpotent for 0 < n < oo, then the Levitzki radical will exist in ~( I ) .

We conclude this paper with a condition which guarantees the Boer radical algebras, that is, the algebras in ~//1, to be locally nilpotent.

(3.1) Lemma. I f A is Boer radical so is any subalgebra B of A Boer radical.

Proof. If B/W is a non-zero homomorphic image of B then we can select an ideal K of A which is maximal with respect to the property that K n B c= 14I. Evidently, K4:A, whence A/K contains a non-zero ideal P/K such that (P/K)2=O. If we put C = W + P n B then B/W will contain the non-zero trivial ideal C/W. Thus B is Boer radical.

(3.2) Lemma. I f B is finitely generated but not nilpotent then B contains an ideal W such that B/W is not nilpotent but B/C is nilpotent for any ideal C of B properly containing 141.

Proof. Let X = {J I J < B and B/J is not nitpotent}, noting that ~t" is non- empty as 0 e X. Let ,/1 __c J2 =c... be a chain in ~ and put J = U J,- We claim

n = l

that J e X. If not, then (B/J) k = 0 for some k. Suppose B is generated by xl . . . . , x~. Then J contains all monomials of degree at least k in the xi's and in particular all monomials of degree s, where k-<.< s < 2k. As there are finitely many such monomials, it follows that some Jm contains all the monomials of degree s, k < s < 2k. In fact, we may show by induction that Jm contains all the monomials of degree at least k. Indeed, if y is a monomial of degree > 2k then as degy > 2, y = ab for some monomials a and b. At least one of a or b has degree at least k, hence is in Jm by the induction hypothesis. As Jm is an ideal, so then is y in Jm" But now we have shown that (B/Jm) k = 0, contrary to the fact that J , is in £r. Therefore, J e 5f.

Having shown that the set ~r is closed when taking chains, we may now use Zorn's lemma to assert that ~r contains a maximal element W, which clearly satisfies the demands of the lemma.

(3.3) Theorem. Let ~( I ) be a variety satisfying the condition (•) ll(A) is nilpotent for each finitely generated nilpotent A e ~lf(I). Then

the Boer radical algebras in ~e~(l) are locally nilpotent.

Proof. Let A be Boer radical and B be a finitely generated subalgebra of A. Assume that B is not nilpotent. According to the previous lemrna, B contains an ideal W which is maximal with respect to the property that B/W is not 3 Math. Ann. 194

34 T. Anderson: The Levitzki Radical in Varieties of Algebras

nilpotent. Certainly, B / W is non-zero, hence from Lemma 3.1, B / W contains a non-zero ideal C / W such that (C/W) 2 =0. From the choice of IV, B/C is nilpotent, as well as finitely generated. Thus II(B/C) is nilpotent. Furthermore, as C and W are ideals of B and C 2 ~_ IV, C / W is a bimodule for the algebra B/C, and as in Lemma 2.4, we may represent B/C in the algebra of linear trans- formations of the space C/W. Using the nilpotence of II(B/C) and the universal mapping property, we conclude there is an integer m such that (C) Ya~'" Yam C= W for all a 1 . . . . . a m ~ B, where Y = R or L. Moreover, as B/C is nitpotent, there is an integer k such that (B) Ybl ... Ybk ~- C for all bl, ..., b k e B. Altogether, for any c I . . . . . Ck+ m in B, (B) Ycl ... Yck+,, = ((B) Yc~ ... Y J Yc~+x "" Yc~+,,

(C) Yck÷~ .-. Y~k÷,,-~ W. Thus, B / W is nilpotent, contrary to the choice of IV. The algebra A was therefore locally nilpotent.

Remark. We note that the condition (~) is really a statement about the structure of the finite dimensional algebras in ~ ( I ) since any finitely generated nilpotent algebra has a finite basis.

References

1. Divinsky, N. : Rings and radicals. Toronto: University of Toronto Press, 1965. 2. Jacobson, N.: Structure and representations of Jordan algebras. Amer. Math. Soc. Colloq.

Publ. 34, Providence (t969). 3. Tsai, C.: Levitzki radical for Jordan rings. Proc. Amer. Math. Soc. 24, 119--123 (1970). 4. Zhevlakov: Solvability and nilpotence of Jordan rings. Algebra i Logika Sem. 5, 37--58 (1966). 5. Zwier, P. : Prime ideals in a class of narings, to appear.

Dr. T. Anderson Department of Mathematics University of British Columbia Vancouver, 8, B.C., Canada

(Received February 15, 1971)