Nil and power-central polynomials in rings
HTML articles powered by AMS MathViewer
- by Uri Leron
- Trans. Amer. Math. Soc. 202 (1975), 97-103
- DOI: https://doi.org/10.1090/S0002-9947-1975-0354764-6
- PDF | Request permission
Abstract:
A polynomial in noncommuting variables is vanishing, nil or central in a ring, $R$, if its value under every substitution from $R$ is 0, nilpotent or a central element of $R$, respectively. THEOREM. If $R$ has no nonvanishing multilinear nil polynomials then neither has the matrix ring ${R_n}$. THEOREM. Let $R$ be a ring satisfying a polynomial identity modulo its nil radical $N$, and let $f$ be a multilinear polynomial. If $f$ is nil in $R$ then $f$ is vanishing in $R/N$. Applied to the polynomial $xy - yx$, this establishes the validity of a conjecture of Herstein’s, in the presence of polynomial identity. THEOREM. Let $m$ be a positive integer and let $F$ be a field containing no $m$th roots of unity other than 1. If $f$ is a multilinear polynomial such that for some $n > 2{f^m}$ is central in ${F_n}$, then $f$ is central in ${F_n}$. This is related to the (non)existence of noncrossed products among ${p^2}$-dimensional central division rings.References
- S. A. Amitsur, The $T$-ideals of the free ring, J. London Math. Soc. 30 (1955), 470–475. MR 71408, DOI 10.1112/jlms/s1-30.4.470
- S. A. Amitsur, A generalization of Hilbert’s Nullstellensatz, Proc. Amer. Math. Soc. 8 (1957), 649–656. MR 87644, DOI 10.1090/S0002-9939-1957-0087644-9
- Edward Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129–132. MR 302689, DOI 10.1016/0021-8693(72)90050-6 I. N. Herstein, Theory of rings, University of Chicago Lectures Notes, 1961.
- Thomas P. Kezlan, Rings in which certain subsets satisfy polynomial identities, Trans. Amer. Math. Soc. 125 (1966), 414–421. MR 217120, DOI 10.1090/S0002-9947-1966-0217120-1 M. Schacher and L. Small, Central polynomials which are $p$th powers, Comm. Algebra (to appear).
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 202 (1975), 97-103
- MSC: Primary 16A38
- DOI: https://doi.org/10.1090/S0002-9947-1975-0354764-6
- MathSciNet review: 0354764