Nil and power-central polynomials in rings
Author:
Uri Leron
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
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Abstract | References | Similar Articles | Additional Information
Abstract: A polynomial in noncommuting variables is vanishing, nil or central in a ring, , if its value under every substitution from
is 0, nilpotent or a central element of
, respectively.
THEOREM. If has no nonvanishing multilinear nil polynomials then neither has the matrix ring
. THEOREM. Let
be a ring satisfying a polynomial identity modulo its nil radical
, and let
be a multilinear polynomial. If
is nil in
then
is vanishing in
. Applied to the polynomial
, this establishes the validity of a conjecture of Herstein's, in the presence of polynomial identity. THEOREM. Let
be a positive integer and let
be a field containing no
th roots of unity other than 1. If
is a multilinear polynomial such that for some
is central in
, then
is central in
.
This is related to the (non)existence of noncrossed products among -dimensional central division rings.
- [1]
S. A. Amitsur, The
-ideals of the free ring, J. London Math. Soc. 30 (1955), 470-475. MR 17, 122. MR 0071408 (17:122c)
- [2] -, A generalization of Hilbert's Nullstellensatz, Proc. Amer. Math. Soc. 8 (1957), 649-656. MR 19, 384. MR 0087644 (19:384a)
- [3] E. Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129-132. MR 46 #1833. MR 0302689 (46:1833)
- [4] I. N. Herstein, Theory of rings, University of Chicago Lectures Notes, 1961.
- [5] T. P. Kezlan, Rings in which certain subsets satisfy polynomial identities, Trans. Amer. Math. Soc. 125 (1966), 414-421. MR 36 #211. MR 0217120 (36:211)
- [6]
M. Schacher and L. Small, Central polynomials which are
th powers, Comm. Algebra (to appear).
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1975-0354764-6
Keywords:
Polynomial identities,
nil polynomials,
power-central polynomials,
Herstein's conjecture,
crossed products
Article copyright:
© Copyright 1975
American Mathematical Society