Prime right alternative rings with commutators in the middle nucleus (Q1178965)

The author studies right alternative rings where commutators are assumed to lie in the middle nucleus. If such a ring is prime, then the ring is either associative, or the middle nucleus is the center of \(R\). His proof is done using the ideal generated by the commutator \([R,M]\) where \(M\) is the middle nucleus and \(R\) is the ring. This ideal lies inside the middle nucleus, and therefore annihiltates the ideal generated by the associators. The author also studies an idempotent \(e\) which is assumed to satisfy \((e,e,R)=0\). If \(R\) is prime and not associative, \(e\) is not zero, and \(e\) is in the middle nucleus, then in fact \(e\) must be the identity element. His proof is based on the first theorem. If \(e\) is in the middle nucleus, then \(e\) is in the center so the idempotent decomposition consists of the end slots which are orthogonal ideals. He gives an example of a six dimensional right alternative algebra with commutators in the middle nucleus which has an idempotent, but which is not associative.