2017/02/04 by Bouchiba, Samir
#13D02 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1702.01345
Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring R. Actually, we translate the theory initiated by A. Grothendieck and R. Sharp and subsequently developed by A. Wadsworth on Krull dimension of tensor products of algebras over a field k into the general setting of algebras over an arbitrary ring R. For this sake, we introduce and study the notion of a fibred AF-ring over a ring R. This concept extends naturally the notion of AF-ring over a field introduced by A. Wadsworth in \citeW to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fibre rings of tensor products of algebras over a ring. Also, given a triplet of rings (R,A,B) consisting of two R-algebras A and B such that A⊗RB≠ \0\, we introduce the inherent notion to (R,A,B) of a B-fibred AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product A⊗RB. As an application, we provide a formula for the Krull dimension of A⊗RB when A and B are R-algebras with A is zero-dimensional as well as for the Krull dimension of A⊗ℤB when A is a fibred AF-ring over the ring of integers ℤ with nonzero characteristic and B is an arbitrary ring. This enables us to answer a question of Jorge Matinez on evaluating the Krull dimension of A⊗ℤB when A is a Boolean ring. Actually, we prove that if A and B are rings such that A⊗ℤB is not trivial and A is a Boolean ring, then dim(A⊗ℤB)=\mbox dim ( \frac B2B ).