On generalizing alternative rings

Rodabaugh, D. J.

doi:10.1090/S0002-9939-1974-0349786-X

On generalizing alternative rings
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by D. J. Rodabaugh PDF

Proc. Amer. Math. Soc. 46 (1974), 157-163 Request permission

Abstract:

Consider a ring $R$ that satisfies the identity $(x,x,x) = 0$ and any two of the three identities: $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x = 0;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z = 0;(w,x \cdot y,z) - x \cdot (w,y,z) - y \cdot (w,x,z) = 0$. In this paper, we prove that if $R$ has characteristic prime to 6 then $R$ semiprime with idempotent $e$ implies $R$ has a Peirce decomposition in which the modules multiply as they do in an alternative ring. If in addition $R$ is prime with idempotent $e \ne 0,1$ then $R$ is alternative.

References

Seyoum Getu and D. J. Rodabaugh, Generalizing alternative rings, Comm. Algebra 2 (1974), 35–81. MR 352195, DOI 10.1080/00927877408822004
Erwin Kleinfeld, Generalization of alternative rings. I, II, J. Algebra 18 (1971), 304-325; ibid. 18 (1971), 326–339. MR 0274545, DOI 10.1016/0021-8693(71)90063-9
Harry F. Smith, Prime generalized alternative rings $I$ with nontrivial idempotent, Proc. Amer. Math. Soc. 39 (1973), 242–246. MR 313348, DOI 10.1090/S0002-9939-1973-0313348-X