Nil-semisimple right alternative algebras

Symplectic alternating nil-algebras

In this paper we continue developing the theory of symplectic alternating algebras that was started in [3]. We focus on nilpotency, solubility and nil-algebras. We show in particular that symplectic alternating nil-2 algebras are always nilpotent and classify all nil-algebras of dimension up to 8.

Small Semisimple Subalgebras of Semisimple Lie Algebras

The goal of Section 2 is to provide a proof of Theorem 2.0.1. Section 3 introduces the necessary facts about Lie algebras and representation theory, with the goal being the proof of Proposition 3.5.7 (ultimately as an application of Theorem 2.0.1), and Proposition 3.3.1. In Section 4 we prove the main theorem, using Propositions 3.3.1 and 3.5.7. In Section 5, we apply the theorem to the special...

Almost semisimple algebras

Notes on semisimple algebras

(1.5) Proposition Let R be a semisimple ring. Then R is isomorphic to a finite direct product ∏s i=1 Ri, where each Ri is a simple ring. (1.6) Proposition Let R be a simple ring. Then there exists a division ring D and a positive integer n such that R ∼= Mn(D). (1.7) Definition Let R be a ring with 1. Define the radical of R to be the intersection of all maximal left ideals of R. The above defi...

Nil, nilpotent and PI-algebras

The notions of nil, nilpotent or PI-rings (= rings satisfying a polynomial identity) play an important role in the ring theory (see e.g. [8], [11], [20]). Banach algebras with these properties have been studied considerably less and the existing results are scattered in literature. The only exception is the work of Krupnik [13], where the Gelfand theory of Banach PI-algebras is presented. Howev...