Primitive Noetherian algebras with big centers (Q2706568)

Let \(R\) be a commutative domain. An \(R\)-module \(M\) is called generically free if there exists \(0\neq s\in R\) such that \(R[s^{-1}]\otimes_RM\) is a free module over the localization \(R[s^{-1}]\). A classical result due to Grothendieck asserts that if \(R\) is Noetherian and \(A\) is a finitely generated commutative \(R\)-algebra, then any finitely generated \(A\)-module is generically free over \(R\). In a recent paper, [J.~Algebra 221, No. 2, 579-610 (1999; Zbl 0958.16024)], \textit{M.~Artin, L.~W.~Small} and \textit{J.~J.~Zhang} have extended this by showing that for a large family of noncommutative algebras any finitely generated module becomes free after localizing at a suitable central element. In contrast, the article under review provides examples of primitive Noetherian algebras \(A\) which are finitely generated over the integers or over algebraic closures of finite fields and which possess a central subring \(R\) such that the faithful simple modules are not generically free over \(R\). The noncommutative version of the Nullstellensatz also fails to hold for these algebras.