Rings with associators in the commutative center
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- by Erwin Kleinfeld PDF
- Proc. Amer. Math. Soc. 104 (1988), 10-12 Request permission
Abstract:
Thedy has introduced the subject of rings which satisfy the identity \[ {\text {(I) }}(R,(R,R,R)) = 0,\] where the commutator is defined by $(a,b) = ab - ba$, and the associator is defined by $(a,b,c) = ab \cdot c - a \cdot bc$ and which satisfy one additional identity such as $(x,x,x) = 0$. Assuming characteristic $\ne 2$ and simplicity, Thedy’s result is that such a ring $R$ must be either commutative or associative. Thedy also showed that simplicity could not be relaxed to prime by presenting some examples which are neither associative nor commutative. We show here that the additional identity assumed by Thedy is in fact unnecessary for we show that if $R$ is simple, characteristic $\ne 2,3$ and satisfies (I) then $R$ must be either commutative or associative.References
- Erwin Kleinfeld, A class of rings which are very nearly associative, Amer. Math. Monthly 93 (1986), no. 9, 720–722. MR 863975, DOI 10.2307/2322290 Armin Thedy, On rings satisfying $((a,b,c),d = 0$, Proc. Amer. Math. Soc. 29 (1971), 213-218.
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 10-12
- MSC: Primary 17A30
- DOI: https://doi.org/10.1090/S0002-9939-1988-0958033-7
- MathSciNet review: 958033