A generalization of commutative and alternative rings. IV
Abstract: We have shown previously for rings of characteristic which satisfy the three identities (i) , (ii) , and (iii) ( , where , and , that under the assumption of no divisors of zero, all such must be either associative or commutative. Here we weaken the Lie-admissible identity (ii) by assuming instead (iv) Lie-admissibility for every subring generated by two elements. It turns out that rings without divisors of zero and of characteristic which satisfy (i), (iii) and (iv) are either commutative or alternative. If is a ring in which every subring generated by two elements is either commutative or associative, then identities (i), (iii) and (iv) hold in , so that this result applies to .
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Keywords: Commutative, alternative, flexible, noncommutative, Jordan, zero divisor
Article copyright: © Copyright 1974 American Mathematical Society