Right alternative algebras and alternators (Q1076790)

The authors study varieties of right alternative algebras \(R\) over fields of characteristic \(\neq 2\), \(\neq 3\). The subspace \(M\) of \(R\) spanned by all alternators \((a,a,b):=(aa)b-a(ab)\) for \(a,b\in R\) is zero iff \(R\) is alternative. The authors systematically study those identities (of degree 4) that force \(M\) to be an ideal. The paper culminates in showing that a right alternative algebra satisfying such an identity is alternative if it is prime, unital and has an idempotent \(e\neq 0\), \(\neq 1\) such that \((e,e,R)=0\) with the possible exception of four identities. The authors leave the study of these exceptions to future work.