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1.
Samir Bouchiba 《代数通讯》2013,41(7):2431-2445
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]. 相似文献
2.
Using the concept of prime submodule defined by Raggi et al. in [16], 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 ∈ 𝔽τ, then cl·K τdim (M) = k τ(M). 相似文献
3.
We describe the structure of a regular semigroup with an associate subgroup the identiy element of which is a mcdial idempotent. As a particular application of this, we obtain the structure of perfect Dubreil-Jacotin semigroups in which the set of residuals of the bimaximum element form a subgroup. 相似文献
4.
Let G be a group acting via ring automorphisms on a commutative unital ring R. If Spec(R) has no infinite antichains and either R a domain or G finitely generated, then R G ? R has the lying-over property. If R is semiquasilocal and dim(R) = 0, then dim(R G ) = 0. If 1 ≤ d ≤ ∞, new examples are given such that d = dim(R) ≠ dim(R G ) < ∞. If G is locally finite on R, then R G ? R satisfies universally going-down. Consequently, if G is locally finite, the S-domain, strong S-domain and universally strong S-domain properties descend from R to R G . If R is a domain, then G is locally finite on R ? R is integral over R G . One cannot delete the “domain” hypothesis. 相似文献
5.
I. Alrasasi 《代数通讯》2013,41(4):1385-1400
Let D be an integral domain with quotient field K. A Bhargava ring over D is defined to be 𝔹 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. 相似文献
6.
Let R be an integral domain. We say that R is a star-domain if R has at least a height one prime ideal and if for each height one prime ideal P of R, R satisfies the acc on P-principal ideals (i.e., ideals of the form aP, a ∈ R). We prove that if R is an APVD with nonzero finite Krull dimension, then the power series ring R[[X]] has finite Krull dimension if and only if R is a residually star-domain (i.e., for each nonmaximal prime ideal P of R, R/P is a star-domain) if and only if R[[X]] is catenarian. 相似文献
7.
Amol Sasane 《Acta Appl Math》2008,103(2):161-168
In this article, we prove that the Krull dimension of several commonly used classes of transfer functions of infinite dimensional
linear control systems is infinite. On the other hand, we also show that the weak Krull dimension of the Hardy algebra
, the disk algebra
and the Wiener algebra
is equal to 1.
A. Sasane is supported by the Nuffield Grant NAL/32420. 相似文献
8.
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. 相似文献
9.
DONG Zhe 《数学年刊B辑(英文版)》2003,24(3):323-330
This paper studies the tensor product RN RM of Jacobson radicals in nest algebras, and obtains that RN RM = {T∈B(H1 H2) : T(N M)(?)N_ M_, N∈N,M∈M}; and based on the characterization of rank-one operators in RN RM,it is proved that if N, M are non-trivial then RN RM=R if and only if N, M are continuous. 相似文献
10.
《Journal of Pure and Applied Algebra》2022,226(4):106887
In the present work, we prove that every class of the divisor class group of a Krull monoid algebra contains infinitely many prime divisors. Several attempts to this result have been made in the literature so far, unfortunately with open gaps. We present a complete proof of this fact. 相似文献
11.
Jan Uliczka 《代数通讯》2013,41(10):3401-3409
In this note we want to generalize some of the results in [1] from polynomial rings in several indeterminates to arbitrary ? n -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*. 相似文献
12.
A. R. Alehaftta 《代数通讯》2018,46(5):2023-2032
We introduce and study the concept of small Krull dimension of a module which is Krull-like dimension extension of the concept of DCC on small submodules. Using this concept we extend some of the basic results for modules with this dimension, which are almost similar to the basic properties of modules with Krull dimension. When for a module A with small Krull dimension, whose Rad(A) is quotient finite dimensional, then these two dimensions for Rad(A) coincide. In particular, we prove that if an R-module A has finite hollow dimension, then A has small Krull dimension if and only if it has Krull dimension. Consequently, we show that if A has properties AB5* and qfd, then A has s.Krull dimension if and only if A has Krull dimension. 相似文献
13.
Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra, ln this paper, we characterize the projectivity (injectivity) of M as a left A#σ H-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σ H, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative. 相似文献
14.
《代数通讯》2013,41(10):4073-4083
Abstract It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ωω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings. 相似文献
15.
利用已知的代数的同调满同态来构造其张量积代数的同调满同态.设A,B,C,D是域k上的有限维代数,如果环同态f:A→C和g:B→D是环的同调满同态,则fg:AB→CD也是环的同调满同态. 相似文献
16.
《代数通讯》2013,41(11):4247-4264
Abstract Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J w (H), for example J w (H) = Ann H ((H/J(H))?n ) for all large enough n. The smallest positive integer n with this property is denoted by l w (H). We prove that l w (H) equals the smallest number n such that (H/J(H))?n contains every projective indecomposable H/J w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p. 相似文献
17.
18.
《Journal of Pure and Applied Algebra》2023,227(2):107179
Let D be an integral domain and let be a torsion-free, ≤-cancellative, subtotally ordered monoid. We show that the generalized power series ring is a Krull domain if and only if D is a Krull domain and S is a Krull monoid. 相似文献
19.
20.
Matthew Ondrus 《代数通讯》2013,41(8):2506-2523
We describe the action of the center of the quantum group Uq () on the tensor product V ? L(λ) of an infinite-dimensional representation V having an infinitesimal character χτ and an irreducible finite-dimensional Uq () representation L(λ) of highest weight λ. We apply this result in order to describe the tensor product of a Whittaker module and a finite-dimensional simple module for the algebra Uq(l2). 相似文献