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1.
Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres. 相似文献
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We introduce the multiplication algebra of a Bernstein algebra, establish its Peirce decomposition relative to an idempotent of A and state some basic properties of this algebra of endomorphtsms 相似文献
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Shushi Harashita 《代数通讯》2013,41(4):1282-1290
In this article, for a noncommutative ring A with some rich structure, we define a ring of Witt vectors with coefficients in A, which is noncommutative unless A is commutative. 相似文献
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In the first correction, induction is not needed (and in factcannot be used). Thus the sentences "We shall use inductionon n,", "by the induction hypothesis" and "The proof still appliesfor n = 1" on lines 9, 7 and 3 from the end, should be omitted. 相似文献
6.
In this paper we investigate the topological localizations of Lie-complete rings. It has been proved that a topological localization of a Lie-complete ring is commutative modulo its topological nilradical. Based on the topological localizations we define a noncommutative affine scheme X = Spf(A) for a Lie-complete ring A. The main result of the paper asserts that the topological localization A(f) of A at f ∈ A is embedded into the ring 𝒪A(Xf) of all sections of the structure sheaf 𝒪A on the principal open set Xf as a dense subring with respect to the weak I1-adic topology, where I1 is the two-sided ideal generated by all commutators in A. The equality A(f) = 𝒪A(Xf) can only be achieved in the case of an NC-complete ring A. 相似文献
7.
A. A. Tuganbaev 《Journal of Mathematical Sciences》2013,191(5):743-747
Comultiplication modules over not necessarily commutative rings are studied. 相似文献
8.
For a commutative ring k,the group G in a G-Galois extension A/ k is uniquely deter-mined by the ring extension A/ k.On the contrary,the Hopf algebra H in an H-Galoisextension A/ k is not unique(see[1 ] ) .In[2 ] ,for a commutative Hopf algebra and acommutative faithfully flat H-Galois extension A/ B(B=Aco H) ,the authors constructedanother commutative Hopf algebra Lsuch that A is also a faithfully flat L-Galois ex-tension.P.Schauenburg generalized H and A to noncommutative cases in[… 相似文献
9.
As G. M. Bergman has pointed out, in the proof of the lemmaon p. 187, we cannot conclude that $$\stackrel{\¯}{S}$$is universal in the sense stated. However, the proof can becompleted as follows: Any element of $$\stackrel{\¯}{S}$$can be obtained as the first component of the solution u ofa system (AI)u+a = 0, (1) where A Sn, a nS and AI has an inverse over L. SinceS is generated by R and k{s}, A can (by the last part of Lemma3.2 of [1]) be taken to be linear in these arguments, say A= A0 + sA1, where A0 Rn, A0 Rn, A1 Kn. Multiplying by (IsA1)1,we reduce this equation to the form (SvBvI)u+a=0, (2) with the same solution u as before, where Bv Rn, sv k{s}1and a nS. Now consider the retraction S k{s} (3) obtained by mapping R 0. If we denote its effect by x x*,then (2) goes over into an equation I.v + a* 0, (4) which clearly has a unique solution v in k{s}; therefore theretraction (3) can be extended to a homomorphism $$\stackrel{\¯}{S}$$ k{s}, again denoted by x x*, provided we can show that u1*does not depend on the equation (1) used to define it. Thisamounts to showing that if an equation (1), or equivalently(2), has the solution u1 = 0, then after retraction we get v1= 0 in (4), i.e. a1* = 0. We shall use induction on n; if u1= 0 in (2), then by leaving out the first row and column ofthe matrix on the left of (2), we have an equation for u2,...,un and by the induction hypothesis, their values after retractionare uniquely determined. Now from (2) we have
where B = (bijv). Applying * and observing that bijvR, we seethat a1 * = 0, as we wished to show. The proof still appliesfor n = 1, so we have a well-defined mapping $$\stackrel{\¯}{S}$$ k{s}, which is a homomorphism. Now the proof of the lemma canbe completed as before. 相似文献
10.
We investigate the prime spectrum of a noncommutative ring and its spectral closure, the extended prime spectrum. We construct
a ring for which the prime spectrum is a spectral space different from the extended prime spectrum and we construct a von
Neumann regular ring for which the prime spectrum is not a spectral space.
The authors are members of the European Research Training Network RAAG (Contract No. HPRN-CT-2001-00271). The first author
was also supported by the Ministry of Education, Science and Sport of Slovenia. 相似文献
11.
Peter Schauenburg 《Applied Categorical Structures》1998,6(2):193-222
It is a key property of bialgebras that their modules have a natural tensor product. More precisely, a bialgebra over k can be characterized as an algebra H whose category of modules is a monoidal category in such a way that the underlying functor to the category of k-vector spaces is monoidal (i.e. preserves tensor products in a coherent way). In the present paper we study a class of algebras whose module categories are also monoidal categories; however, the underlying functor to the category of k-vector spaces fails to be monoidal. Instead, there is a suitable underlying functor to the category of B-bimodules over a k-algebra B which is monoidal with respect to the tensor product over B. In other words, we study algebras L such that for two L-modules V and W there is a natural tensor product, which is the tensor product VBW over another k-algebra B, equipped with an L-module structure defined via some kind of comultiplication of L. We show that this property is characteristic for ×B-bialgebras as studied by Sweedler (for commutative B) and Takeuchi. Our motivating example arises when H is a Hopf algebra and A an H-Galois extension of B. In this situation, one can construct an algebra L:=L(A,H), which was previously shown to be a Hopf algebra if B=k. We show that there is a structure theorem for relative Hopf bimodules in the form of a category equivalence
. The category on the left hand side has a natural structure of monoidal category (with the tensor product over A) which induces the structure of a monoidal category on the right hand side. The ×B-bialgebra structure of L that corresponds to this monoidal structure generalizes the Hopf algebra structure on L(A,H) known for B=k. We prove several other structure theorems involving L=L(A,H) in the form of category equivalences
. 相似文献
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13.
The l-quasi Nil Radical of Lattice-ordered Rings 总被引:1,自引:0,他引:1
§ 1.l- q Ideals and the l- q Radical of an l- ring Generalizing the notion of quasi nil of rings to lattice-ordered rings(l-rings) ,we de-fine l-quasi nil ideals(l-q ideals) and the l-quasi nil radical(l-q radical) of l-rings.Theproperties of the l-q radical and the structure of l-q semisimple rings are studied.Fur-thermore,we consider the l-q radical of full l-matrix l-rings and l-rings with minimumcondition on single sided l-ideals. Throughout this paper R denotes an l-ring if not spec… 相似文献
14.
由环的P-性质所确定的根 总被引:1,自引:0,他引:1
定义了环的P根P、弱拟P根Pw 和拟P根PQ,证明它们均为Am itsurKurosh 根且P= Pw 为特殊根,给出了P半单环的结构定理和P根的模刻划 相似文献
15.
Supersemiprime Rings and Supersemiprime Radical 总被引:4,自引:1,他引:3
§1. IntroductionInthisrecentyears,theprimeringswithspecialpropertieshavebeenstudiedsomeau-thors,forexample,N.T.Groenewaldintroducedthecompletelyprimeringsandthecompletelyrimeradicalin[1].D.HandelmanandJ.Lawrencediscussedthestronglyprimeringsandthestronglyprimeradicalin[2].S.Veldsmanstudiedthesuperprimeringsandthesuperprimeradicalin[3].ZhangXianjunintroducedtheAbsolutelysemiprimeringsandtheabsolutelysemiprimeradicalandsoon.Theaimofthispaperistostudythesemiprimeringswithspecialpropertiesan… 相似文献
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Byungdo Kim 《数学学报(英文版)》2000,16(1):21-28
Abstract
Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A→A such that [D(x), x]D(x)[D(x),x] ∈ rad(A) for all x∈A. In this case, D(A) ⊆ rad (A).
The author has been supported by Kangnung National University, Research Fund, 1998 相似文献
19.
With the help of functional integrals in Fock space, under some analytical assumptions, we construct representations for exponents of quadratic functions of creation and annihilation operators with noncommuting coefficients. 相似文献
20.
文中给出了Radicaltotal环上投射模的分解定理. 对一个 ~uniform 维数有限的 Totalfree 环 R, 该文证明 R 是一个总体维数≤ 1 的诺特环, 且 R上的任何投射模必同构于$ \bigoplus\limits_{i\in I}Re_{i}$, 其中每个 $e_{i}$ 均为 $R$ 的非零幂等元. 此外, 文中还给出了一些相关的例子. 相似文献