A Wedderburn theorem for alternative algebras with identity over commutative rings

Abstract:

In this paper, we study alternative algebras ${\mathbf {\Lambda }}$ over a commutative, associative ring R with identity. When ${\mathbf {\Lambda }}$ is finitely generated as an R-module, we define the radical J of ${\mathbf {\Lambda }}$. We show that matrix units and split Cayley algebras can be lifted from ${\mathbf {\Lambda }}/J$ to ${\mathbf {\Lambda }}$ when R is a Hensel ring. We also prove the following Wedderburn theorem: Let ${\mathbf {\Lambda }}$ be an alternative algebra over a complete local ring R of equal characteristic. Suppose ${\mathbf {\Lambda }}$ is finitely generated as an R-module, and ${\mathbf {\Lambda }}/J$ is separable over $\bar R$ ($\bar R$ the residue class field of R). Then there exists an $\bar R$-subalgebra S of ${\mathbf {\Lambda }}$ such that $S + J = {\mathbf {\Lambda }}$ and $S \cap J = 0$.