Characterization of non-matrix varieties of associative algebras

Abstract

A variety of associative algebras is called a non-matrix variety if it does not contain the algebra of 2 × 2 matrices over the base field K. There are some known characterizations of non-matrix varieties. We give some new characterizations in terms of properties of nilelements. Let V be a variety of associative algebras over an infinite field. Then the following conditions are equivalent: (1) V is a non-matrix variety, (2) any finitely generated algebra A ∈ V satisfies an identity of the form [x 1 , x 2 ] · · · [x 2s−1 , x 2s ] ≡ 0, (3) let A ∈ V; then for any nilelements a, b ∈ A, the element a + b is again a nilelement. Let E be the Grassmann algebra in countable many generators. We also give similar characterizations for non-matrix varieties over fields of characteristic zero that do not contain E or E ⊗ E.

Non-matrix polynomial identity enveloping algebras

“…Amitsur had already proved in [Am] that the Jacobson radical of a relativelyfree algebra of countable rank is nil and Samoilov in [Sam] proved that the Jacobson radical of a relatively free algebra of countable rank over an infinite field of positive characteristic is a nilideal of bounded index. The non-matrix varieties have been further studied in [BRT,MPR,R97].…”

Multiplicatively collapsing and rewritable algebras

“…In fact, the only possibility is is nilpotent when A is finitely generated (see [2] or [12], for example). Because the last statement depends only on the K-variety satisfied by p n = 0, the nilpotence index of C(A) is bounded by a function depending only on m, n, and possibly K. To see there is a bound independent of K, let t 0 be the nilpotence index of C(A) when A is the relatively-free (countably generated) algebra Q-algebra satisfying p n = 0.…”

Lie solvable enveloping algebras of characteristic two

“…Indeed, if char F = 2 then M 2 (F) is Lie center-by-metabelian. The non-matrix varieties of algebras have been extensively studied, see for example [10,12,13,20], and enveloping algebras have received special attention in this respect [3,4,23]. Using the standard PI-theory, like Posner's Theorem, one can deduce that if R is an associative algebra that satisfies a non-matrix PI over a field F of characteristic p then [R, R]R is nil.…”