## Algebra & Number Theory

### Inner derivations of alternative algebras over commutative rings

#### 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
Revised: 26 September 2008
Accepted: 26 October 2008
First available in Project Euclid: 20 December 2017

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

#### 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