Santos González

COORDINATE SETS OF GENERALIZED GALOIS RINGS

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to , the so-called Teichmuller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop is right (or left) primitive.

Classification of bernstein algebras of type (3, n - 3)

A classification of Bernstein algebras in dimensions n ⩽ 4 has been made by Holgate in [2], however that article contains no classification up to isomorphism, the problem is solved by Lyubich in [4] when K = R or C, and by Cortes [1] in the general case. Also Lyubich has given in [5] a classification of the regular nonexceptional Bernstein algebra of type (3,n−3) and a classification but not up to isomorphism of nonregular nonexceptional Bernstein algebras of type (3,n − 3) when K = C. The aim of this paper it to characterize, up to isomorphism, Bernstein algebras of type(2, n − 2) and nonexceptional of type(3, n −3) over a infinite commutative field K whose characteristic is different from 2.

Order relation in Jordan rings and a structure theorem

It is shown that the relation sg defined by x < y if and only if xy = x2, x2y = xy2 = x} is an order relation for a class of Jordan rings and we prove that a Jordan ring R is isomorphic to a direct product of Jordan division rings if and only if < is a partial order on R such that R is hyperatomic and orthogonally complete. Introduction. The usual relation in Boolean rings is extended to reduced rings A (has no nilpotent elements) when it is expressed as a *s b if and only if ab = a2 (Abian [1,2] and Chacron [4]) and it is proved that any ring R equipped with the relation < is isomorphic to a direct product of division rings if and only if < is an order relation in R such that R is hyperatomic and orthogonally complete. In [5] Myung and Jimenez extend the above results to any alternative ring and they show that the same results do not hold for Jordan rings, because the ring Q of real quaternions under the product a ■ b = \{ab + ba) becomes a Jordan ring Q + without nonzero nilpotent elements, but the relation < is not a partial order on Q+. Also, Q+ is a Jordan division ring, in the sense that Ua = 2R2a Ra2 is invertible on Q+ for every a # 0 in Q+, where Ra is the right multiplication in Q + by a. The aim of this paper is to define an order relation for Jordan rings and to obtain a structure theorem for these rings. 1. Preliminaries. In this section we give the elementary properties and definitions related to Jordan rings which will be needed in the paper. A Jordan ring is a commutative nonassociative ring R satisfying (xy)x2 = x{yx2) for all x, y e R. In terms of the associator (x, y, z) = (xy)z — x(yz) this is to say (x, y, x2) = 0. If A is an associative ring then it is well known that the elements of A form a Jordan ring under the same operation of addition and under the new multiplication a • b = \{ab + ba) where ab denote the associative product of a and b in A. This ring will be denoted by A+. It is easy to see that the powers of an element in A are the same under the Jordan product as under the associative product, more generally any subring of a Jordan ring which is generated by one Received by the editors June 11, 1985 and, in revised form, October 28, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 17C10.

GROUP CODES OVER NON-ABELIAN GROUPS

Let G be a finite group and F a field. We show that all G-codes over F are abelian if the order of G is less than 24, but for F = ℤ5 and G = S4 there exist non-abelian G-codes over F, answering to an open problem posed in [J. J. Bernal, A. del Rio and J. J. Simon, An intrinsical description of group codes, Des. Codes Cryptogr.51(3) (2009) 289–300]. This problem is related to the decomposability of a group as the product of two abelian subgroups. We consider this problem in the case of p-groups, finding the minimal order for which all p-groups of such order are decomposable. Finally, we study if the fact that all G-codes are abelian remains true when the base field is changed.

Cyclic Generalized Galois Rings

ABSTRACT The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S, S* = S\ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = ℤ/nℤ with n = 4 or n = p r for some odd prime p. Without associativity, S* is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.

On nilpotency of the barideal of a bernstein algebra

It is given a new bound for the index of nilpotency of the barideal of a finited generated Bernstein algebra. Also, it is found the index of solvability of the barideal of a Bernstein algebra and a bound of nilpotency for the square of the barideal.

Quasiisomorphisms of bernstein algebras

In this paper the notion of quasiisomorphism of Bernstein algebras is introduced and studied. The aim of the paper is to get a classification of Bernstein algebras more general than the given by isomorphisms. In the last part of the paper the classification, up to quasiisomorphisms, of Bernstein algebras of dimensions 3 and 4 is given and compared with the corresponding classification up to isomorphism.

On regular bernstein algebras

Abstract In this article we prove a matricial equivalence to the classification of regular Bernstein algebras. On the other hand, we describe all regular nuclear Bernstein algebras with stochastic realization of type ( n + 1, 1).

Baric, bernstein and jordan algebras

The aim of this paper is the study of relations between some special classes of baric algebras, as Jordan algebras, Bernstein or second order Bernstein algebras, train algebras and power-associative algebras. It will be proved that the train equation of a Bernstein algebra that is train of rank r is unique, what leads to a characterization of these algebras similar to the well known characterization of a Jordan-Bernstein algebra in terms of a Peirce decomposition.

Order relation in quadratic Jordan rings and a structure theorem

It is shown that the relation defined by x < y if and only if Vxx = Vxy and UTx = Uxy = Uvx is an order relation for quadratic Jordan algebras without nilpotent elements, which extends our previous one for linear Jordan algebras, and reduces to the usual Abian order for associative algebras. We prove that a quadratic Jordan algebra is isomorphic to a direct product of division algebras if and only if the algebra has no nilpotent elements and is hyperatomic and orthogonally complete. Preliminaries. Throughout this paper,