Generalized Weyl algebras are tensor Krull minimal

Formulae for calculating the Krull dimension of noetherian rings obtained b y the authors and their collaborators are used to calculate Krull dimension for certain classes of algebras. An F-algebra T is said to be tenser Krull minimal (TKM) with respect to a class of F-algebras Omega if R(T circle ti mes B) = R(T) + R(B), for each B epsilon Omega. We show that generalized We yl algebras over affine commutative F-algebras, where F is an uncountable a lgebraically closed field, are TKM with respect to the class of countably g enerated left noetherian F-algebras. This simplifies the task of calculatin g many Krull dimensions. In addition, we develop an improved formula fur th e Krull dimension of a skew Laurent extension D[x, x(-1); sigma], where D i s a polynomial algebra over an algebraically closed field, and sigma is an affine automorphism. Finally, we calculate the Krull dimension of the noeth erian down-up algebras introduced by Benkart. (C) 2001 Academic Press.