On Krull and Valuative Dimension of Tensor Products of Algebras

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 dim _v (A? _k B) in the case where A ? B.     

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.     

S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Let 𝒜 = (A _n ) _n≥0 be an ascending chain of commutative rings with identity, S ? A ₀ a multiplicative set of A ₀, and let 𝒜[X] (respectively, 𝒜[[X]]) be the ring of polynomials (respectively, power series) with coefficient of degree i in A _i for each i ∈ ?. In this paper, we give necessary and sufficient conditions for the rings 𝒜[X] and 𝒜[[X]] to be S ? Noetherian.     

AF-Domains and Their Generalizations

In this article, we are concerned with the study of the dimension theory of tensor products of algebras over a field k. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any k-algebra A such that the polynomial ring in one variable A[X] is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of A? _k B for any k-algebra A such that A[X] is an AF-domain and any k-algebra B generalizing the main theorem of Wadsworth in [16 Wadsworth , A. R. ( 1979 ). The Krull dimension of tensor products of commutative algebras over a field . J. London Math. Soc. 19 : 391 – 401 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Down-Up Algebras From Trees

It is shown that the global dimension of any n-ary down-up algebra A _n  = A(n,α, β,γ) is less than or equal to n + 2, and when γ _i  = 0 for all i (A _n is graded by total degree in the generators), then the global dimension of A _n is n + 2. Furthermore, a sufficient condition for A _n to be prime is given; when γ _i  = 0 for all i this condition is also necessary. An example is given to show that the condition is not always necessary.     

Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p ₁ ? p ₂ ? ··· ? p _n be a chain of nonmaximal prime ideals of T such that T _{p n} is a regular local ring. We construct a chain of excellent local domains A _n  ? A _n?1 ? ··· ? A ₁ such that for each 1 ≤ i ≤ n, the completion of A _i is T, the generic formal fiber of A _i is local with maximal ideal p _i , and if I is a nonzero ideal of A _i then A _i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht _T Q, then there is a chain of excellent local domains B ₀ ? B ₁ ? ··· ? B _h  ? T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B _i ) = i, the completion of B _i is isomorphic to T[[X ₁, X ₂,…, X _i ]] where the X _j 's are indeterminants, and the formal fiber of Q ∩ B _i is local.     

On Partial Tilting Complexes

Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ^? ∈ 𝒦^b(𝒫 _A ) with Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i > 0. We will provide a sufficient condition for P ^? to be a direct summand of a silting complex. Also, in case Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ^? to be a direct summand of a tilting complex.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

Quasi-Armendariz Rings Relative to a Monoid

A ring R is called an M-quasi-Armendariz ring (a quasi-Armendariz ring relative to a monoid M) if whenever elements α = a ₁ g ₁ + a ₂ g ₂ + ··· + a _n g _n , β = b ₁ h ₁ + b ₂ h ₂ + ··· + b _m h _m  ? R[M] satisfy α R[M]β = 0, then a _i Rb _j  = 0 for each i, j. After discussing some basic properties of M-quasi-Armendariz rings, we consider the influence of transformation of the monoid M and the ring R on this property. Particularly, we give some sufficient conditions for the monoids M, N, and the ring R under which R is M × N-quasi-Armendariz if and only if R is M-quasi-Armendariz and N-quasi-Armendariz.     

On the Centralizers of Derivations with Engel Conditions

Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([d(x), x] _n ) = 0 for all x ∈ R, then char R = 2, d ² = 0, and δ = αd, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[d(x), x] _n  | x ∈ R} in R coincides with the center of R.     

Quotient Rings of Ore Extensions with More than One Indeterminate

Let R be a domain and R[X; D] the Ore extension of R by a sequence D of derivations of R. If D has length ≥ 2, we show that the symmetric Utumi quotient ring of R[X; D] is U _s (R)[X; D], where U _s (R) is the symmetric Utumi quotient ring of R. Consequently, X-inner automorphisms of R[X; D] are induced by units of U _s (R) and the extended centroid of R[X; D] consists of those elements α in the center of U _s (R) such that δ(α) = 0 for all δ ? D. These extend the known results for free algebras.     

Krull Dimension and Classical Krull Dimension of Modules

Using the concept of prime submodule defined by Raggi et al. in [16 Raggi , F. , Rios , J. , Rincón , H. , Fernández-Alonso , R. , Signoret , C. ( 2005 ). Prime and irreducible preradicals . J. Algebra Appl. 4 ( 4 ): 451 – 466 .[Crossref], [Web of Science ®] , [Google Scholar]], for M ∈ R-Mod we define the concept of classical Krull dimension relative to a hereditary torsion theory τ ∈M-tors. We prove that if M is progenerator in σ[M], τ ∈M-tors such that M has τ-Krull dimension then cl.K _τdim (M) ≤ k _τ(M). Also we show that if M is noetherian, τ-fully bounded, progenerator of σ[M], and M ∈ 𝔽_τ, then cl·K _τdim (M) = k _τ(M).     

On Clean Rings

A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A₁ ⊕ A₂ with M′ ? M, there is a decomposition M′ =M₁ ⊕ M₂ such that A = M′ ⊕ [A₁ ∩ (M₁ ⊕ B)] ⊕ [A₂ ∩ (M₂ ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A _k k[x]_(x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]_(x).     

Radicals of Skew Polynomial and Skew Laurent Polynomial Rings Over Skew Armendariz Rings

In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x ^?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N ₀(R[x; α]) = Ni?_*(R)[x; α] and J(R[x, x ^?1; α]) = N ₀(R[x, x ^?1; α]) = Ni?_*(R)[x, x ^?1; α].     

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ?: T → K: = T/M the natural surjection, and R the pullback ?^?1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x ₁]]…[x _n ]] to be catenarian, where each [x _i ]] is fixed as either [x _i ] or [[x _i ]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x ₁]]…[x _n ]]) → Spec(k[x ₁]]…[x _n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x ₁]]…[x _n ]] is catenarian.