## Algebra & Number Theory

- Algebra Number Theory
- Volume 2, Number 8 (2008), 927-968.

### Inner derivations of alternative algebras over commutative rings

Ottmar Loos, Holger Petersson, and Michel Racine

#### Abstract

We define Lie multiplication derivations of an arbitrary non-associative algebra $A$ over any commutative ring and, following an approach due to K. McCrimmon, describe them completely if $A$ is alternative. Using this description, we propose a new definition of inner derivations for alternative algebras, among which Schafer’s standard derivations and McCrimmon’s associator derivations occupy a special place, the latter being particularly useful to resolve difficulties in characteristic $3$. We also show that octonion algebras over any commutative ring have only associator derivations.

#### Article information

**Source**

Algebra Number Theory, Volume 2, Number 8 (2008), 927-968.

**Dates**

Received: 6 April 2008

Revised: 26 September 2008

Accepted: 26 October 2008

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ant/1513805230

**Digital Object Identifier**

doi:10.2140/ant.2008.2.927

**Mathematical Reviews number (MathSciNet)**

MR2457357

**Zentralblatt MATH identifier**

1191.17011

**Subjects**

Primary: 17D05: Alternative rings

Secondary: 17A36: Automorphisms, derivations, other operators 17A45: Quadratic algebras (but not quadratic Jordan algebras) 17B40: Automorphisms, derivations, other operators

**Keywords**

inner derivations alternative algebras derivation functors composition algebras automorphisms

#### Citation

Loos, Ottmar; Petersson, Holger; Racine, Michel. Inner derivations of alternative algebras over commutative rings. Algebra Number Theory 2 (2008), no. 8, 927--968. doi:10.2140/ant.2008.2.927. https://projecteuclid.org/euclid.ant/1513805230