## Abstract

The following result related to a question of G. Bergman and L. Small [J. Algebra, 33 (1975), 435-462] is obtained:. Let R be an associative algebra over a field F, 1 ε{lunate} R, char F ≠ 2, 5, such that R satisfies the standard polynomial identity S_{2n} and for every maximal ideal M in R, R M doesn't satisfy all the identities of (n - 3) × (n - 3) matrices. Then R is Azumaya. We also obtain an improvement of a theorem of M. Artin and C. Procesi, we show:. Let R be a ring with a unit such that R satisfies S_{2n} and R M doesn't satisfy S_{2k}, k < n, for every maximal ideal M in R. Then R is Azumaya of constant rank n^{2}. We also show that to answer Bergman and Small's question affirmatively for given n, it suffices to check the non-vanishing of a polynomial S_{2n}^{*} on the matrix ring M_{k}(F), for k > n 2.

Original language | English |
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Pages (from-to) | 24-45 |

Number of pages | 22 |

Journal | Journal of Algebra |

Volume | 89 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1984 |

## ASJC Scopus subject areas

- Algebra and Number Theory