Narrow Results

The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor–Bendixson rank of any member of this space. For instance, the Cantor–Bendixson rank of the free commutative ring on n generators is ωⁿ, where ω is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.

We describe the two-generated limits of abelian-by-(infinite cyclic) groups in the space of marked groups using number theoretic methods. We also discuss universal equivalence of these limits.

This work is devoted to interpretation of concepts of Zariski dimension of an algebraic variety over a field and of Krull dimension of a coordinate ring in algebraic geometry over algebraic structures of an arbitrary signature. Proposed dimensions are ordinal numbers (ordinals).

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonnier's Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules.

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring

Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim(R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim(R) = dim(A[[X]]) = dim(A) + 1.

Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is a direct sum of a projective module and a module Q, where Q is either injective or noetherian. This provides an affirmative answer to a question raised by P. F. Smith.

We give a constructive approach to the well known classical theorem saying that an integral extension doesn't change the Krull dimension.

This paper is concerned with the study of the dimension theory of tensor products of algebras over a field k. We answer an open problem set in [6] and compute dim(A ⊗_k B) when A is a k-algebra arising from a specific pullback construction involving AF-domains and B is an arbitrary k-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback A is in fact a generalized AF-domain.

The purpose of this paper is to explore new aspects of the prime ideal structure of tensor products of algebras over a field k. We prove that given a k-algebra A and a normal field extension K of k (in the sense of Galois theory), then for any prime ideals P₁ and P₂ of K ⊗_k A lying over a fixed prime ideal p of A, there exists a k-automorphism σ of K such that (σ ⊗_kid_A)(P₁) = P₂. As an Application, we establish a result related to the dimension theory of tensor products stating that, for two arbitrary k-algebras A and B, the minimal prime ideals of p ⊗_k B + Aσ_k q have the same height, for any prime ideals p and q of A and B, respectively.

Let R be a commutative regular ring of dimension d, I be an ideal of R and M be a non-zero finitely generated R-module. In this paper we show that Huneke's two different conjectures are equivalent. Also we provide some partially affirmative answers to them. In fact it is shown that the Bass numbers of are finite for all i ≥ 0, whenever d ≤ 3. Also if (R, m) be regular local ring, we show that the Bass numbers are finite, for all i ≥ 0 and all j ≥ 0, and , for all i ≥ 0 and all j ≥ 0, whenever height(I) = 1.

In this paper, we introduce and study the notion of G-type domains (a domain R is G-type if its quotient field is countably generated R-algebra). We extend some of the basic properties of G-domains to G-type domains. It's observed that a prime ideal of R[x₁, x₂,…,x_n,…] is G-type if and only if its contractions in R, R[x₁, x₂,…,x_n] for all n ≥ 1 are G-type. Using this concept we give a natural proof of the well-known Hilbert Nullstellensatz in infinite countable-dimensional spaces. Characterizations of Noetherian G-type domains, Noetherian G-type domains with the countable prime avoidance property are given. As a consequence, we observe that in complete Noetherian semi-local rings, G-type ideals and G-ideals are the same. Rings with countable Noetherian dimension which are direct sum of G-type domains are fully determined. Finally, we characterize Noetherian rings in which G-type ideals are maximal.

We establish several characterizations of maximal non-Prüfer and maximal non-integrally closed subrings of a field. Special attention is given when finiteness conditions are satisfied. Several numerical characterizations are then obtained. The paper is concluded with examples that exhibit the obtained results.

Let R ⊆ S be a unital extension of commutative rings, with the integral closure of R in S, such that there exists a finite maximal chain of rings from R to S. Then S is a P-extension of R, is a normal pair, each intermediate ring of R ⊆ S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many -subalgebras of S. Each chain of rings from R to S is finite if dim(R) = 0; or if R is a Noetherian (integral) domain and S is contained in the quotient field of R; or if R is a one-dimensional domain and S is contained in the quotient field of R; but not necessarily if dim(R) = 2 and S is contained in the quotient field of R. Additional domain-theoretic applications are given.

In this paper, we characterize maximal non-ACCP subrings R of a domain S in case (R, S) is a residually algebraic pair and R is semilocal. In this paper, we also consider a K-algebra S, a nonzero proper ideal I of S and a subring D of the field K and we determine necessary and sufficient conditions in order that D + I is a maximal non-ACCP subring of S. This gives an example of a maximal non-ACCP subring R of a domain S such that (R, S) is a normal pair and R is not semilocal.

We study a class of down–up algebras 𝒜(α, β, ϕ) defined over a polynomial base ring 𝕂[t₁,…,t_n] and establish several analogous results. We first construct a 𝕂-basis for the algebra 𝒜(α, β, ϕ). As an application, we completely determine the center of 𝒜(α, β, ϕ) when char 𝕂 = 0, and prove that the Gelfand–Kirillov dimension of 𝒜(α, β, ϕ) is n + 3. Then, we prove that 𝒜(α, β, ϕ) is a noetherian domain if and only if β ≠ 0, and 𝒜(α, β, ϕ) is Auslander-regular when β ≠ 0. We show that the global dimension of 𝒜(α, β, ϕ) is n + 3, and 𝒜(α, β, ϕ) is a prime ring except when α = β = ϕ = 0. Finally, we obtain some results on the Krull dimensions, isomorphisms and automorphisms of 𝒜(α, β, ϕ).

Let $(R,𝔪)$ be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and $I$ an ideal of $R$. Let $M$ be a nonzero finitely generated $R$-module and $i\ge 0$ be an integer. In this paper we show that, the $R$-module $H_{𝔪}^i(M)$ is nonzero and $I$-cofinite if and only if $Rad(I+0{:}_R{H}_{𝔪}^i(M))=𝔪$. Also, several applications of this result will be included.

An $n$-dimensional quantum torus is defined as the $F$-algebra generated by variables $x_1,\dots ,x_n$ together with their inverses satisfying the relations $x_i{x}_j=q_{ij}{x}_j{x}_i$, where $q_{ij}\in F$. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of $〈x_1,\dots ,x_n〉$ that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

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 Grothendieck and Sharp and subsequently developed by 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 fibered AF-ring over a ring $R$. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc.19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber 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 $A{\otimes }_{R}B\ne \{0\}$, we introduce the inherent notion to $(R,A,B)$ of a $B$-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product $A{\otimes }_{R}B$. As an application, we provide a formula for the Krull dimension of $A{\otimes }_{R}B$ when either $R$ or $A$ is zero-dimensional as well as for the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a fibered AF-ring over the ring of integers $\mathbb{Z}$ with nonzero characteristic and $B$ is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a Boolean ring. Actually, we prove that if $A$ and $B$ are rings such that $A{\otimes }_{\mathbb{Z}}B$ is not trivial and $A$ is a Boolean ring, then dim$(A{\otimes }_{\mathbb{Z}}B)=\text{dim}(\frac{B}{2B})$.