A Wedderburn decomposition for certain generalized right alternative algebras

Smith, Harry F.

doi:10.1090/S0002-9939-1976-0419540-0

A Wedderburn decomposition for certain generalized right alternative algebras
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Proc. Amer. Math. Soc. 58 (1976), 1-7 Request permission

Abstract:

Finite-dimensional nonassociative algebras are considered which satisfy certain subsets of the following identities: (1) $(x,x,x) = 0$, (2) $(wx,y,z) + (w,x,[y,z]) = w(x,y,z) + (w,y,z)x$, (3) $(w,x \cdot y,z) = x \cdot (w,y,z) + y \cdot (w,x,z)$, (4)$(x,y,z) + (y,z,x) + (z,x,y) = 0$. It is first observed that nil algebras satisfying (1) and (2) are solvable. The standard Wedderburn principal theorem is then established both for algebras satisfying (1), (2) and (3) and for algebras which satisfy (2) and (4). Throughout it is assumed that the base fields have characteristic different from 2 and 3.

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