Erwin Kleinfeld

Generalization of alternative rings, I

where we define (a, b, c) = (ab)c a(bc), and (a, b) = ab ba. It is well known that (i) is true in a right alternative ring of characteristic different from two. But being an identity of degree four it is of course weaker than the right alternative identity. Then (ii) is the counterpart of (i), and a consequence of the left alternative identity. A number of generalizations of the alternative identities have been considered, some of which are listed in the reference section. Some of these have taken on the form of identities which are shared by commutative and hence Jordan rings, while others are shared by Lie rings. The object of this generalization is to ascertain whether there exist interesting examples of rings like the Cayley numbers, hence our choice of (i) and (ii). Throughout our rings will be assumed to have characteristic different from two and three, meaning in this case that there exist no elements of additive order two and three. The main results are the following. A necessary and sufficient condition for a ring of characteristic different from two and three, satisfying (i) and (ii) to be alternative is that whenever there exist elements a, b, c which are contained in a subring which can be generated by two elements and (a, B, c)” = 0, then (a, 6, c) = 0. So therefore a ring without nilpotent

On centers of alternative algebras

Shestakov showed that in an arbitrary alternative ring of characteristic ≠ 2, the fourth power of every associator Is in the commutative center. He raised the question whether this might bp so for the square of every associator. The answer to this question is no, Which is demonsuapared by an exumple of an altor-native algobre of dimension 107.

The structure of generalized accessible rings

BY ERWIN KLEINFELD, MARGARET HUMM KLEINFELD AND J. FRANK ROSIER Communicated by I. N. Herstein, October 9, 1968 In [3] Schaf er has defined generalized standard rings as rings satisfying the identities (1) (%,y,*)=0 (2) (x, y, z)x + (y, z, x)x + (2, x, y)x = (xf y, xz) + {y, xz, x) + (xz, x} y) (3) (x, y, wz) + (w, y, xz) + (z, y} xw) = (x, (w, z, y)) + (x, w, (y, z)) and observed that these identities imply (4) (y,y, (x,z)) = 0, and for characteristic not three (5) (x,y,x>)=0. Schaf er determines the structure of simple, finite-dimensional generalized standard algebras of characteristic not 2 or 3 by showing that they must be either commutative Jordan or alternative. Previously one of the authors [2] had studied accessible ringst which are defined by the identities (6) (x, y, z) + (0, x, y) (x, z, y) = 0 and (7) ((w,x),y,z)=0. The structure of accessible rings is determined in that paper as it turns out that an accessible prime ring must be either associative or commutative. Both of these results generalize some results on standard algebras by Albert [ l] . In the present announcement we define an even more general class of rings called generalized accessible, as those satisfying the identities (8) (^(2,y,y)) = o,

A classification of

By complicated arguments involving the notion of quasi-equivalence, Albert showed tha t the principal algebras of this type were the (7, ô) algebras. Subsequent investigations into the s t ructure of (7, 8) algebras were made by Kleinfeld, Kokoris, Maneri , Hentzel and others, and there exists a satisfactory s t ructure theory for these algebras (see [6; 7; 8; 9; 10; 11; 12; 13; 14]). The reason we return to the classification of algebras satisfying (1.2) and (1.3) is twofold. First of all, in view of the fact t ha t associator dependent algebras have already been classified [12], we may restrict ourselves to those algebras which are not associator dependent , thereby avoiding lengthy calculations involving quasi-equivalence. Secondly, condition (1.4) eliminates from the very beginning any consideration of Lie algebras and so is too restrictive. Ideally a survey of 2-varieties should explain which of these varieties are of known type, for instance Lie or al ternat ive, and which are uninterest ing, so t h a t future investigations can concentrate on the rest. In view of the large number of

2

Albert is an interactive computer system for building nonassociative algebras [2]. In this paper, we suggest certain techniques for using Albert that allow one to posit and test hypotheses effectively. This process provides a fast way to achieve new results, and interacts nicely with traditional methods. We demonstrate the methodology by proving that any semiprime ring, having characteristic ≠2,3, and satisfying the identities (a, b, c) - (a, c, b) = (a, [b, c]d) = 0,is asociative. This generalizes a recent result by Y. Paul [7].

-varieties

Generalizations of theorems on simple Novikov algebras by E. I. Zel’manov and J. M. Osborn to a subvariety in the join of associative and Novikov are obtained.

Rings with (a, b, c) = (a, c, b) and (a, [b, c]d) = 0: a case study using albert

In rings of characteristic not two, (1) implies (2) [2], while in rings of characteristic two this is not the case [3]. In the following note we establish that in the free strongly right alternative ring R generated by a and b we have ((a, b), a, b) O0. We also know from previous work [2, Lemma 5(i)] that ((a, b), a, b)2 = 0 in R. While it was already known from theoretical considerations that the free right alternative ring of characteristic not two on two or more generators would have to have at least one element x HZ 0 such that x2 = 0, since otherwise the Main Theorem of [2 ] implies it would have to be alternative, which we know is not the case, this enables one to locate such an element specifically for the first time. In order to prove that ((a, b), a, b)0 in R, it suffices to give an example of a strongly right alternative algebra in which this is not an identity. Our example has basis elements ql, , qg. The only nonzero products of basis elements are the following twelve: q2= q3, qlq2 2 =q5, q2ql = q6, q2=q4, q2q7 = q9, q3q2 =-q7, q4ql =-q8, q4q3 =-q9, q5ql = q7, q6q2==qs, q6q5= q9, and q8ql=q9. From this information one may construct the multiplication table for the basis elements of the algebra. To see that the algebra A thus defined is right alternative, it is sufficient to verify (1) whenever x, y, and z are basis elements. The identity must then hold for all elements because of the linearity of the associator. It can easily be verified that q7, q8, and qg are in the nucleus and so (1) is automatically satisfied if any of these appear. It can also be verified that qiR R = 0 = qjR2 for i = 3, 5. Hence (1) holds if x = q3 or

A Generalization of Novikov Rings

Abstract Let R be a ring of characteristic not two, satisfying the following three identities, which are consequences of the right alternative identity: (x, x, x) = 0, (wx, y, z) + (w, x, (y, z)) = w(x, y, z) + (w, y, z)x, and (y, y, x)k = 0, for some positive integer k = k(x, y). A simple ring in this class which is not associative is alternative if and only if R has an idempotent e such that there are no nilpotent elements in R1(e) and R0(e), the zero and one subspaces of the Albert decomposition. This generalizes the comparable result for right alternative rings by Humm-Kleinfeld.

A nonidentity for right alternative rings

Let A be a right alternative algebra, and [ A, A ] be the linear span of all commutators in A . If [ A, A ] is contained in the left nucleus of A , then left nilpotence implies nilpotence. If [ A, A ] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [ A, A ] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A , then A has a Wedderburn decomposition

Generalization of right alternative rings

Publisher Summary This chapter discusses the application of computer in mathematical problem, which is to determine whether there exist non-Desarguesian projective planes of order 16. It presents the calculation of Veblen–Wedderburn systems of a certain type. If it is tried to be done in the crudest possible way, the computations get too cumbersome, and so it was necessary to devise a clever scheme to reduce the problem to manageable proportions. The computer printed out all the examples, for it indicated that the problem was solved and that there had to be at least one non-Desarguesian plane. Moreover, the computer is used concerning an identity in alternative rings. A ring may be defined to be alternative in case every subring, which can be generated by two elements is associative.