A nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset X of R we have xXy ≠ 0 whenever 0 ≠ x, y ∈ R. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n = 1. This paper is devoted primarily to an investigation of uniform bounds of primeness in matrix rings over fields. It is shown that the existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the n × n matrix ring over F.

The Shirshov-Cohn theorem asserts that in a Jordan algebra (with 1), any subalgebra generated by two elements (and 1) is special. Let J be a Jordan algebra with 1, a, b elements of J and let a1, a2, …, an be invertible elements of J such that

Where

are Jordan polynomials. In [2, p. 425] Jacobson conjectured that for any choice of the Pi the subalgebra of J generated by 1, a, b, a1…, an is special.

This note continues the development of the infinite-dimensional genetic algebra approach to problems of population genetics. Two algebras are studied. One describes the familiar problem of a quantitative characteristic, and the other provides a way of treating the whole chromosome as an entity.

Abstract. We consider the structure of the Kac modules V(Λ) for dominant integral doubly atypical weights Λ of the Lie superalgebra s1(2/2). Primitive vectors of V(Λ) are constructed and it is shown that the number of composition factors of V(Λ) for such Λ is in exact agreement with the conjectures of [HKV]. These results are used to show that the extended Kac-Weyl character formula which was proved in [VHKTl] for singly atypical simple modules of s1(m/n), and conjectured to be valid for all finite dimensional irreducible representations of sl(m/n) in [VHKT2] is in fact valid for all finite-dimensional doubly atypical simple modules of s1(2/2).

This paper gives variants of results from classical algebraic geometry and commutative algebra for quadratic algebras with conjugation. Quadratic algebras are essentially two-dimensional algebras with identity over commutative rings with identity, on which a natural operation of conjugation may be defined. We define the ring of conjugate polynomials over a quadratic algebra, and define c-varieties. In certain cases a close correspondence between standard varieties and c-varieties is demonstrated, and we establish a correspondence between conjugate and standard polynomials, which leads to variants of the Hilbert Nullstellensatz if the commutativering is an algebraically closed field. These results may be applied to automated Euclidean geometry theorem proving.

Lie algebras whose finite-dimensional modules decompose into direct sums of modules involving only one type of irreducible are investigated. Some vanishing theorems for the cohomology of some infinite-dimensional Lie algebras are thereby obtained.

We investigate the identities which hold in the associated Lie rings of groups of prime exponent. The multilinear identities which hold in these Lie rings are known, and it is conjectured that all the identities which hold in these Lie rings are consequences of multilinear ones. This is known to be the case for the associated Lie rings of two generator groups of exponent 5, and we provide some additional avidence for the conjecture by confirming that it also holds true for the associated Lie rings of three generator groups of exponent five.

We consider certain affine Kac-Moody Lie algebras. We give a Lie theoretic interpretation of the generalized Euler identities by showing that they are associated with certain filtrations of the basic representations of these algebras. In the case when the algebras have prime rank, we also give algebraic proofs of the corresponding identities.

The classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

Two local nilpotent properties of an associative or alternative ring A containing an idempotent are shown. First, if A = A11 + A10 + A01 + A00 is the Peirce decomposition of A relative to e then if a is associative or semiprime alternative and 3-torsion free then any locally nilpotent ideal B of Aii generates a locally nilpotent ideal 〈B〉 of A. As a consequence L(Aii) = Aii ∩ L(A) for the Levitzki radical L. Also bounds are given for the index of nilpotency of any finitely generated subring of 〈B〉. Second, if A(x) denotes a homotope of A then L(A) ⊆ L(A(x)) and, in particular, if A(x) is an isotope of A then L(A) = L(A(x)).

It is proved that a regular essentially closed and weakly homomorphically closed proper subclass of rings consists of semiprime rings. A regular class M defines a supernilpotent upper radical if and only if M consists of semiprime rings and the essential cover Mk of M is contained in the semisimple class S U M. A regular essentially closed class M containing all semisimple prime rings, defines a special upper radical if and only if M satisfies condition (S): every M-ring is a subdirect sum of prime M-rings. Thus we obtained a characterization of semisimple classes of special radicals; a subclas S of rings is the semisimple class of a special radical if and only if S is regular, subdirectly closed, essentially closed, and satisfies condition (S). The results are valid for alternative rings too.

The Brown-McCoy radical is the upper radical defined by the class of simple rings with identities. For associative or alternative rings the Brown-McCoy radical is hereditary, and its semi-simple class consists of all subdirect products of simple rings with identities. In this paper we present some classes of simple non-associative algebras whose upper radicals behave similarly. Classifications are then obtained of ‘most’ semi-simple radical classes of (γ, δ) and right alternative rings.

In addition to the results of the paper (Bachturin (1974)) we give the precise form of the necessary and sufficient conditions ensuring that all irreducible representations of a Lie algebra were of finite bounded degree.

The conjugacy of Cartan subalgebras of a Lie algebra L over an algebraically closed field under the connected automorphism group G of L is inherited by those G-stable ideals B for which B/Ci is restrictable for some hypercenter Ci of B. Concequently, if L is a restrictable Lie algebra such that L/Ci restrictable for some hypercenter Ci of L, and if the Lie algebra of Aut L contains ad L, then the Cartan subalgebras of L are conjugate under G. (The techniques here apply in particular to Lie algebras of characteristic 0 and classical Lie algebras, showing how the conjugacy of Cartan subgroups of algebraic groups leads quickly in these cases to the conjugacy of Cartan subalgebras.)

An example is constructed of a locally finite variety of non-associative algebras which satisfies the maximal condition on subvarieties but not the minimal condition. Based on this, counterexamples to various conjectures concerning varieties generated by finite algebras are constructed. The possibility of finding a locally finite variety of algebras which satisfies the minimal condition on subvarieties but not the maximal is also investigated.