On Quotient Rings in Alternative Rings

We introduce a notion of left nonsingularity for alternative rings and prove that an alternative ring is left nonsingular if and only if every essential left ideal is dense, if and only if its maximal left quotient ring is von Neumann regular (a Johnson-like Theorem). Finally, we obtain a Gabriel-like Theorem for alternative rings.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Q_cl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Q_l(R) of an arbitrary ring R is proved, i.e., Q_l(R) = S₀(R)^?1R where S₀(R) is the largest left regular denominator set of R. It is proved that Q_l(Q_l(R)) = Q_l(R); the ring Q_l(R) is semisimple iff Q_cl(R) exists and is semisimple; moreover, if the ring Q_l(R) is left Artinian, then Q_cl(R) exists and Q_l(R) = Q_cl(R). The group of units Q_l(R)* of Q_l(R) is equal to the set {s^?1t | s, t ∈ S₀(R)} and S₀(R) = R ∩ Q_l(R)*. If there exists a finitely generated flat left R-module which is not projective, then Q_l(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).     

Morita Invariance in Cyclic Homology for Nonunital Algebras

Let E and F be two linear subspaces of an algebra A over a field of characteristic zero and let A(EF), A(FE) be the subalgebras of A generated by EF and FE, respectively. We show that A(EF) and A(FE) have the same periodic cyclic homology.     

The Maximal Graded Left Quotient Algebra of a Graded Algebra1)

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.     

Morita Context环的幂零性

设(A,B,V,W,(),[])是一个Morita Context,C=A VW B是对应的Morita Context环.用基本环论方法,给出了C与A,B,V,W之间关于环的诣零性,幂零性,局部幂零性,N—诣零性,P—性等性质的关系.     

关于Quotient Context of Morita Context的讨论

本文讨论了ＱｕｏｔｉｅｎｔＣｏｎｔｅｘｔｏｆ Ｍｏｒｉｔａ Ｃｏｎｔｅｘｔ的一些性质,并在此基础上证明：如果ＤＲ是一个完备Ｇａｂｒｉｅｌ拓扑,则一个（Ｐ,Ｑ）的商（Ｐ,Ｑ）作为Ｒ与Ｓ之间ＭＣ,它的商同构于（Ｐ,Ｑ）的一个商。     

Morita Context环的若干性质

研究由Morita Contexts所确定的环,讨论它的K-好环性质,弱Semicom-mutative性质,有许多单式正则元性质和广义稳定环性质等.     

Gorenstein-Projective Modules over Morita Rings

Let △(φ,ψ) =(A BMA ANBB) be a Morita ring which is an Artin algebra.In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring △(φ,ψ) and the algebras A and B.We prove that if △(φ,ψ) is a Gorenstein algebra and both MA and AN (resp.,both NB and BM) have finite projective dimension,then A (resp.,B) is a Gorenstein algebra.We also discuss when the CM-freeness and the CM-finiteness of a Morita ring △(φ,ψ) is inherited by the algebras A and B.     

带Morita对偶的偏序集环

设Ｒ是有１的结合环，Ｉ是任意偏序集，ＲＩ是Ｒ上Ｉ的偏序集环．本文考虑了带对偶的偏序集环，得到：ＲＩ带Ｍｏｒｉｔａ对偶当且仅当Ｒ带Ｍｏｒｉｔａ对偶．推广了已有的在Ｒ是有限偏序集时的有关结果     

Rings of the Morita Contexts and Semihereditary Maximal V-Orders

Some necessary and sufficient conditions are given for rings of the Morita contexts to be semihereditary maximal V-orders, Prüfer rings and Dubrovin valuation rings.     

Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

It is proved that the filter dimension is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand–Kirillov dimension for all nonzero finitely generated A-modules M. In fact, a stronger result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. A direct consequence of this fact is that an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H _c for integral c: for all nonzero finitely generated H _c -modules M. For these class of algebras, it gives an affirmative answer to a question of Ken Brown about symplectic reflection algebras. Presented by Alain Verschoren.     

有足够幂等元的群分次环上的Morita对偶

首先定义了具有足够幂等元的群分次环的Ｓｍ ａｓｈ 积并讨论其性质,其次,借助Ｓｍ ａｓｈ 积给出了这类分环上的分次Ｍｏｒｉｔａ对偶的刻划．     

群分次环和Morita等价

设Ｒ是Ｇ－分次环，Ｈ是Ｇ的子群，本文通过构造两个Ｍｏｒｉｔａｃｏｎｔｅｘｔ，给出了环Ｂ｛Ｈ｝Ｍｏｒｉｔａ等价于Ｒ＿Ｈ，Ｒ＃Ｇ￣＊Ｍｏｒｉｔａ等价于Ｒ＿Ｈ＃Ｈ￣＊的充分必要条件．     

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

Quotient rings and Fountain-Gould left orders by the local approach

We study Fountain-Gould left orders in semiprime rings coinciding with their socles by means of local rings at elements.     

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.     

正则QB环上的Morita Contexts

假设T是Morita Contexts(A,B,M,N,φ,φ)的环,如果A和B是正则QB环,本文证明了T中任何元素都是幂等元和伪逆元之和.     

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ? T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.     

Maximal Inequalities and an Invariance Principle for a Class of Weakly Dependent Random Variables

The aim of this paper is to investigate the properties of the maximum of partial sums for a class of weakly dependent random variables which includes the instantaneous filters of a Gaussian sequence having a positive continuous spectral density. The results are used to obtain an invariance principle for strongly mixing sequences of random variables in the absence of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences.