# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## On a generalization of alternative and Lie ringsHTML articles powered by AMS MathViewer

by Erwin Kleinfeld
Trans. Amer. Math. Soc. 155 (1971), 385-395 Request permission

## Abstract:

Alternative as well as Lie rings satisfy all of the following four identities: (i) \$({x^2},y,z) = x(x,y,z) + (x,y,z)x\$, (ii) \$(x,{y^2},z) = y(x,y,z) + (x,y,z)y\$, (iii) \$(x,y,{z^2}) = z(x,y,z) + (x,y,z)z\$, (iv) \$(x,x,x) = 0\$, where the associator \$(a,b,c)\$ is defined by \$(a,b,c) = (ab)c - a(bc)\$. If \$R\$ is a ring of characteristic different from two and satisfies (iv) and any two of the first three identities, then it is shown that a necessary and sufficient condition for \$R\$ to be alternative is that whenever \$a,b,c\$ are contained in a subring \$S\$ of \$R\$ which can be generated by two elements and whenever \${(a,b,c)^2} = 0\$, then \$(a,b,c) = 0\$.
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