Isomorphically rigid algebras

Abstract

Various classes of non-associative algebras possessing the property of being rigid under abstract isomorphisms are studied.In the paper, we continue the research on rigidity of algebras initiated in [1]. In [2], the property of being rigid for algebras under automorphisms was explored (see also [3]). Here, we look at such properties for algebras under abstract isomorphisms.Below, we deal with three classes of non-associative algebras. It is proved that each of these classes is rigid given a suitable concept of being rigid for an algebra under abstract isomorphisms. Of prime concern in the argument is a class of finite-dimensional algebras over scalar fields. We couch the concept of representative scalar fields for such algebras and prove a modular generalization of the known Myasnikov-Remeslennikov theorem in [4]. THEOREM 1. Let V K and W L be finite-dimensional indecomposable algebras with non-zero multiplication over representative scalar fields K and L. In V K and W L , we can then define invariant series of ideals such aswhich satisfy the following:(1) if φ : V → W is an isomorphism of the algebras treated as rings, then n = m, and for any i = 1, . . . , n,(2) there is a field isomorphism σ : K → L such that (αx) φ ∈ α σ x φ + W i+1 for any x ∈ V i and any α ∈ K. PRELIMINARIESThe starting point for studying into rigidity of finite-dimensional algebras is the following fact (see [1, Cor. 1]).Let V be a finite-dimensional commutative local ring. Then V has a unique maximal field of finite dimensionality if and only if either V is a field, or some field of finite dimensionality for V is a field of * Supported by RFBR grant No. 06-01-00159a. P/B 410, Novosibirsk, 630090, Russia; ponom@online.sinor.ru.