Multilinear identities of the matrix ring

Author:
Uri Leron

Journal:
Trans. Amer. Math. Soc. **183** (1973), 175-202

MSC:
Primary 16A42

MathSciNet review:
0332873

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Abstract: Let *V* be a vector space over a field *F* of zero characteristic, which is acted upon by the symmetric group. Systems of generators for *V* are constructed, which have special symmetry and skew symmetry properties. This is applied to prove that every multilinear polynomial identity of degree which holds in the matrix ring is a consequence of the standard identity . The notions of rigid and semirigid sequences of matrices are defined and treated.

**[1]**A. S. Amitsur and J. Levitzki,*Minimal identities for algebras*, Proc. Amer. Math. Soc.**1**(1950), 449–463. MR**0036751**, 10.1090/S0002-9939-1950-0036751-9**[2]**I. N. Herstein,*Noncommutative rings*, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR**0227205****[3]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0332873-3

Keywords:
Matrix ring,
multilinear polynomial,
standard identity,
rigid and semirigid sequences

Article copyright:
© Copyright 1973
American Mathematical Society