120 ARCH. MATH.

The Wedderburn-Mal'cev theorems in a locally finite setting

By L ~ STrW~T

The methods developed recently [6, 7, 8] to study a class of locally finite Lie algebras analogous to FC-groups apply mutatis mutandis to a similar class of as- sociative algebras. Since this leads to a generalization of the Wedderburn structure theorems and the conjugacy theorem of Mal'cev [4] a brief account, avoiding side- issues and special considerations relevant to the Lie algebra case, may be of indepen- dent interest. Other generalizations of these thoorems to infinite dimensions have been given by Curtis [2], Reisel [5], and Zelinsky [9]: these require the radical J to satisfy

~ j t --_ O, i = l

and the whole algebra to be complete in the J-adie topology (defined by the powers of J). Conditions such as that the quotient by J be of finite or countable dimension are also usually imposed. Our results follow a somewhat different pattern.

Let 1 be a field. An associative 1-algebra will be called ideally finite ff it is generated by finite-dimensional ideals. Using theorem 1 of Jaeobson [3], p. 10, it is easy to verify that the nilpotent, nil, Jaeobson, and Levitzld radicals coincide: they are all equal to the sum of the radicals of the finite-dimensional ideals. This common value we call the radical. An algebra with zero radical we call semisimple (or semiprimitive). By a set, arable algebra we shall mean a (possibly infinite) direct sum of central simple algebras: this is equivalent to the usual definition (Albert [1], p. 44) in finite dimensions. :Note tha t ideally finite algebras are locally finite: the converse is of course false.

We shall prove the following generalization of the theorems of Wedderburn and Mal'eev:

Theorem. Let A be an ideally finite associative balgebra with radical J. Then

(a) A/J is semisimple and is the direct sum o] finite-dlmensional simple ideals. I]/urther A /J is separable then

(b) J is complemented in A by a (semisimple) subalgebra C. I / 1 is algebraically closed then (e) The set o/complements to J is permuted transitively by the automorphism group

o/A.

Vol. XXu 1976 The Wedderburn-Mal'cev theorems 121

P r o o f . I t is well known that X/J is semisimple. So let B be any semisimple ideally flnlte algebra, generated by a system {Ft}i~i of finite-dimensional ideals. The nil- potent radical of any Ft is an ideal of B, so must be zero. Hence each Fl is a direct sum of simple ideals. I t is easy to see tha t these must also be ideals of B, so tha t B is generated by finite-dimensional simple ideals. I t follows in the usual way (cf. [6], lemma 4.4, p. 88) tha t B is a dire~ sum of finite-dimensional simple ideals. This proves (a).

For future use note tha t this implies tha t every ideal of B is complemented in B.

Nex~ we prove (b). Let C be a maximal separable semisimple subalgebra of A, which exists by Zorn's lemma. We claim tha t J- t - C = A. Since C is semisimple this implies o r r3 C = 0, whence (b) holds. I t remains to prove J-}- C ~ A : we assume for a contradiction tha t J - i -C ~ A. Then there exists a finite-dimensional ideal F of A such tha t F ~ J ~- C. I f J ' is the radical o f f then J ' = Jr~F (Jacobson [3], theorem 1, p. 10) and so F-t- C 4= o r' -}- C. Let A' = F-l- C, and let K be the annihilator (two-sided) of F in A'. Since F is finite-dimensional K has finite eodimension in A', from which it follows that C'----C r3 K has finite codimension in C. Also C' is an ideal of C, so there is a complement C" such that C = U' @ C". Since C is separable so are C' and C". Now A' splits as a direct sum

A ' = ( F + C " ) |

Further C" is maximal semisimple separable in F + g" , for if g " ' is larger then C' annihilates C"' so tha t C"' ~ C ' is separable semisimple and larger than C. The finite-dimensional structure theory now implies that F-t- C" ~ J' ~ C". But now

F ~ C - ~ F ~ C " - ~ C ' - ~ J ' ~ C " ~ C ' ~ J ' ~ C

which is a contradiction. This proves (b). Note that a similar proof works for Lie algebras, and could be used to simplify

that given in [6, 7] for the existence of Levi factors.

l~inally we prove (c). Let C and C" be complements to J in A. I t is easy to see, arguing as in theorem 4.1 of [8] and using the fmite-climensional structure theory, that for each finite-dimensional ideal F~ of A the subalgebras

complement J~, the radical of Fi . Let

By the conjugacy theorem of Mal'cev [4] it follows that ~r =~ 0. We define a partial order on the indexing set I for the fmite-rllmensional ideals Ft as follows: i ~ ] if and only if F~CFI. Then I is directed. For i ~ j there is a map

obtained by restriction; and we obviously have a projective limit system

Now Aut (Ft) is an affiue algebraic group over ~, and ~r is a coset in this of the sub-

122 I. STEWART ARCH. MATH.

group stabilizing Ci, which is also algebraic. The maps ~j~ are induced by algebraic group morphisms. Since ~ is algebraically closed we m a y apply theorem 2.3 o f [8] (the hypothesis of characteristic zero is unnecessary there) to conclude tha t

= proj lira d i =~ 0.

I f we take ~ = (:r ~ ~4 then we can make :r ac t as an au tomorphism of A by defining its action on F~ to be tha t of ~ : the condit ion on project ive limits makes sure this action is well defined. Then obviously C ~ = C" and (e) is proved.

Unlike [2, 5, 9] the au tomorphism ~ is no t asserted to be inner (or quasi-inner): the example o f an infinite direct sum of finite-dimensional algebras in which comple- ments to the radical are no t unique shows at once t h a t it need no t be. I t is ' locally ' inner in a similar sense to t h a t of [8], a t least provided the definition of the ~4~ is sui tably chosen: in part icular :r m a y be chosen to fix all the ideals of A.

A c k n o w l e d g e m e n t . This work was done at the Univers i ty of Tfibingen and financed by a Forsehungsst ipendium from the Alexander yon Humboldt -St i f tung, Bonn-Bad Godesberg.

Relerences

[1] A. A. ALBERT, Structure of algebras. Providence 1939. [2] C. W. CURTIS, The structure of non-semisimple algebras. Duke Math. J. 21, 79--85 (1954). [3] N. JACOBSON, Structure of rings. Providence 1956 (revised 1964). [4] A. I. M_~'cEv, On the representation of an algebra as a direct sum of the radical and a semi-

simple subalgebra. Dokl. Akad. Nauk. SSSR 36, 42--45 (1942). [5] R. B. R]~Is~L, A generalization of the Wedderburn-Mal'cev theorem to infinite-dimensional

algebras. Proc. Amer. Math. Soc. 7, 493--499 (1956). [6] I. N. STEWART, Structure theorems for a class of locally finite Lie algebras. Proc. London

Math. Soc. (3) 24, 79--100 (1972). [7] I. N. STV, Wa_nT, Levi factors of infinite-dimensional Lie algebras. J. London Math. Soc. (2) 5,

488 (1972). [8] L N. STV, W~a~T, Conjugacy theorems for a class of locally finite Lie algebras. Compositio

Math. 80, i81--210 (1975). [9] D. Z~n~sKx', Raising idempotents. Duke Math. J. 21, 315--322 (1954).

Eingegangen am3. 3. 1975

Anschrift des Autors: Ian Stewart Mathematics Institute University of Warwick Coventry, England