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The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's [7] constructive characterization of the Krull dimension. While Lombardi's characterization uses the lexicographic monomial order, ours uses the graded (reverse) lexicographic order or, in fact, any graded rational monomial order. Apart from this, the paper contains some related results and some examples which readers may find illuminating. [ABSTRACT FROM AUTHOR]

AF-rings are algebras over a field k which satisfy the Altitude Formula over k. This paper surveys a few works in the literature on the Krull and valuative dimensions of tensor products of AF-rings. The first section extends Wadsworth's classical results on the Krull dimension of AF-domains to the larger class of AF-rings. It also provides formulas for computing the valuative dimension with effect on the transfer of the (locally) Jaffard property. The second section studies tensor products of AF-rings over a zero-dimensional ring. Most results on algebras over a field are extended to these general constructions. The third section establishes formulas for the Krull and valuative dimensions of tensor products of pullbacks issued from AF-domains. Throughout, examples are provided to illustrate the scope and limits of the results. [ABSTRACT FROM AUTHOR]

In this paper we study the class of w-Jaffard domains in pullback constructions, and give new examples of these domains. In particular we give examples to show that the two classes of w-Jaffard and Jaffard domains are incomparable. As another application, we establish that for each pair of positive integers (n, m) with n + 1 ≤ m ≤ 2n + 1, there is an (integrally closed) integral domain R such that w-dim(R) = n and w[X]-dim(R[X]) = m. [ABSTRACT FROM AUTHOR]

Let D be an integral domain with quotient field K. A Bhargava ring over D is defined to be x(D): = {f ∈ K[X] | ∀ a ∈ D, f(xX + a) ∈ D[X]}, where x ∈ D. A Bhargava ring over D is a subring of the ring of integer-valued polynomials over D. In this article, we study the prime ideal structure and calculate the Krull and valuative dimension of Bhargava rings over a general domain D. [ABSTRACT FROM AUTHOR]

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type ⋆[ X] on the polynomial ring D[ X], such that, if n := ⋆-dim( D), then n+1 ≤ ⋆[ X]-dim( D[ X]) ≤ 2 n+1. We also establish that if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain, then ⋆[ X]-dim( D[ X]) = ⋆- dim( D)+1. Moreover we define the semistar valuative dimension of the domain D, denoted by ⋆-dim v ( D), to be the maximal rank of the ⋆-valuation overrings of D. We show that ⋆-dim v ( D) = n if and only if ⋆[ X 1, . . . , X n ]-dim( D[ X 1, . . . , X n ]) = 2 n, and that if ⋆-dim v ( D) < ∞ then ⋆[ X]-dim v ( D[ X]) = ⋆-dim v ( D) + 1. In general ⋆-dim( D) ≤ ⋆-dim v ( D) and equality holds if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain. We define the ⋆-Jaffard domains as domains D such that ⋆-dim( D) < ∞ and ⋆-dim( D) = ⋆-dim v ( D). As an application, ⋆-quasi-Prüfer domains are characterized as domains D such that each (⋆, ⋆′)-linked overring T of D, is a ⋆′-Jaffard domain, where ⋆′ is a stable semistar operation of finite type on T. As a consequence of this result we obtain that a Krull domain D, must be a w D -Jaffard domain. [ABSTRACT FROM AUTHOR]

This article is concerned with the dimension theory of tensor products of algebras over a field k . In fact, we provide formulas for the Krull and valuative dimension of A ⊗k B when A and B are k -algebras such that the polynomial ring A [n] is an AF-domain for some positive integer n . Also, we compute dimv (A ⊗k B ) in the case where A ⊆ B . [ABSTRACT FROM AUTHOR]

A multiplicative subset of a commutative ring contains the zero element precisely if the set in question meets every prime ideal. While this form of Krull's Lemma takes recourse to transfinite reasoning, it has recently allowed for a crucial reduction to the integral case in Kemper and the third author's novel characterization of the valuative dimension. We present a dynamical solution by which transfinite reasoning can be avoided, and illustrate this constructive method with concrete examples. We further give a combinatorial explanation by relating the Zariski lattice to a certain inductively generated class of finite binary trees. In particular, we make explicit the computational content of Krull's Lemma. [ABSTRACT FROM AUTHOR]