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Some properties of the Levitzki radical in alternative rings

  • Michael Rich (a1)

Abstract

Two local nilpotent properties of an associative or alternative ring A containing an idempotent are shown. First, if A = A11 + A10 + A01 + A00 is the Peirce decomposition of A relative to e then if a is associative or semiprime alternative and 3-torsion free then any locally nilpotent ideal B of Aii generates a locally nilpotent ideal 〈B〉 of A. As a consequence L(Aii) = AiiL(A) for the Levitzki radical L. Also bounds are given for the index of nilpotency of any finitely generated subring of 〈B〉. Second, if A(x) denotes a homotope of A then L(A)L(A(x)) and, in particular, if A(x) is an isotope of A then L(A) = L(A(x)).

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References

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[1] Albert, A. A., (1952), ‘On simple alternative rings’, Canad. J. Math. 4, 129135.
[2] Anderson, T., (1974), ‘A note on strong radicals’, Acta Math. Acad. Sci. Hungar. 25, 56.
[3] Erickson, T. S., ‘The Jordan prime radical in alternative algebras’, unpublished.
[4] Herstein, I. N., (1968), ‘Noncommutative rings’, (The Math. Assoc. of Amer.).
[5] Kaplansky, I., (1951), ‘Semi-simple alternative rings’, Portugal. Math. 10, 3750.
[6] McCrimmon, K., (1971), ‘A characterization of the Jacobson-Smiley Radical’, J. Algebra 18, 565573.
[7] McCrimmon, K., (1973), ‘Alternative algebras satisfying polynomial identities’, J. Algebra 24, 283292.
[8] Rich, M., (1980), ‘Nilpotent ideals in alternative rings’, Bull. Canad. Math. Soc. 23, 299303.
[9] Slater, M., (1968), ‘Ideals in semiprime alternative rings’, J. Algebra 8, 6076.
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