A Wedderburn theorem for alternative algebras with identity over commutative rings

Abstract

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