15 AF-Rings and Locally Jaffard Rings

FLORIDA GIROLAMI Dipartimento di Matematica, Universita di Roma “La Sapienza”, Piazzale A. Moro 5, 00185 Roma, Italy; e-mail GIROLAMI@ITCASPUR.bitnet.

0 Introduction

AF-rings have been introduced by A.R. Wadsworth in [W] for computing the (Krull) dimension of the tensor product of A1 and A2, dim(A1kA2), where A1 and A2 are commutative algebras over a field k. Previously R.Y.Sharp proved in [S] that if K1 and K2 are extension fields of k, then

dim(K1kK2)=min{t.d.(K1:k)),t.d.(K2:k)}.

Since, for each prime ideal P of a k-algebra A, ht(P)+ t.d(A/P:k) t.d(AP:k) (cf.[ZS, p.10] and [W, p.392]), for giving a generalization of Sharp’s result, A.R. Wadsworth has studied the k-algebras A which satisfy the ...

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