Nil and power-central polynomials in rings

Author:
Uri Leron

Journal:
Trans. Amer. Math. Soc. **202** (1975), 97-103

MSC:
Primary 16A38

MathSciNet review:
0354764

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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