Primitive noetherian algebras with big centers

Irving, Ronald

doi:10.1090/S0002-9939-00-05809-3

Primitive noetherian algebras with big centers
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by Ronald S. Irving PDF

Proc. Amer. Math. Soc. 129 (2001), 1587-1593 Request permission

Abstract:

Recent work of Artin, Small, and Zhang extends Grothendieck’s classical commutative algebra result on generic freeness to a large family of non-commutative algebras. Over such an algebra, any finitely-generated module becomes free after localization at a suitable central element. In this paper, a construction is given of primitive noetherian algebras, finitely generated over the integers or over algebraic closures of finite fields, such that the faithful, simple modules don’t satisfy such a freeness condition. These algebras also fail to satisfy a non-commutative version of the Nullstellensatz.

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