关于半交换环与强正则环

本文得到了环Ｒ是强正则环的若干充分必要条件，证明了下面条件是等价的：（１）Ｒ是强正则的；（２）Ｒ是半交换正则的；（３）Ｒ是半交换的左ＳＦ－环；（４）Ｒ是半交换的ＥＬＴ环，且使得每个单左Ｒ－模是Ｐ－内射的或者平坦的；（５）Ｒ是半交换右非奇异的左ＳＦ－环；（６）Ｒ是半素的半交换左（或右）Ｐ－内射环．     

一类环上HX环的结构

自李洪兴１９９１年提出了ＨＸ环以来，人们一直有这么一个问题没解决，就是是否存在非平凡的ＨＸ环的例子？但至今既没找到非平凡的ＨＸ环，也没有证明任一环Ｒ仅存在平凡的ＨＸ环。针对这个问题，本文提出并证明了一类环仅有平凡ＨＸ环，还给出了一系列的结构定理。这样，既为证明任一环Ｒ仅有平凡的ＨＸ环的猜想有新的启示，也为人们指明无须在这一类环上寻找非平凡ＨＸ环。     

无限矩阵环和完备环

无限矩阵环和完备环郭善良（复旦大学）Ｓｈａｎｎｙ在１９７１证明了一个环Ｒ是半单Ａｒｔｉｎ环当且仅当Ｒ上的无限矩阵环是ＶｏｎＮｅｕｍａｎｎ正则环［１］。这也就是说一个环的无限矩阵环在一定程度上唯一确定了Ｒ本身。我们注意到若ＥｎｄＦ，为ＶｏｎＮｅｕｍｍａ...     

半值环的两个交换结果

定理１设Ｒ是半值环，ｎ为固定的正整数，如果Ｒ满足条件：存在依赖于(?)_ｘ，ｙ的两个字ｋ（Ｘ，Ｙ），ｔ（Ｘ，Ｙ），其中｜ｋ｜_X＞１，｜ｔ｜_X＝１，｜ｋ｜_Y≥｜ｔ｜_Y，｜ｔ｜_Y≤ｎ，使ｋ（ｘ，ｙ）－ｔ（ｘ，ｙ）∈Ｉ（Ｒ），则Ｒ是交换环。定理２设Ｒ是半值环，如果Ｒ满足条件：存在正整数ｍ＝ｍ（ｘ，ｙ）＞１，ｎ＝ｎ（ｙ），使得（ｘ^ｎｙ）^m－ｘ     

关于C—环

主要讨论了C－环的结构以及它与其它环类之间的重要关系,从而较深刻地刻划这种环。     

关于Buchsbaum环的注记

本文证明了下述结论，设Ａ是一个级数为ｄ的Ｂｕｃｈｓｂａｕｍ环，（ａ₁，ａ₂，…，ａ_n）是Ａ的一个参数系统，则任何正整数ｎ，Ａ／（ａ₁，ａ₂，…，ａ_k）ⁿ（１≤ｋ≤ｄ）仍是ｄ－ｋ维的Ｂｕｃｈｓｂａｕｍ环．     

The Rings Characterized by Minimal Left Ideals

We study these rings with every minimal left ideal being a projective, direct summand and a p-injective module, respectively. Some characterizations of these rings are given, and the relations among them are obtained. With these rings, we characterize seinisiinple rings. Finally, we introduce MC2 rings, and give some characterizations of MC2 rings.     

Ideals in tensor powers of the enveloping algebra u(sl 2)

Let U = U(_sl₂)^?n be the tensor power of n copies of the enveloping algebra U(sl ₂) over an arbitrary field K of characteristic zero. In this paper we list the prime ideals of U by generators and classify them by height. If Z is the center of U and J is a prime ideal of Z, there are exactly 2₅ prime ideals I of U with I ∩ Z = J, where 0 ≤ s = s(J) ≤ n is an integer. Indeed, with respect to inclusion, they form a lattice isornorphic to the lattice of subsets of a set. When J is a maximal ideal of Z, there are only finitely many two-sided ideals of U containing J, They are presented by generators and their lattice is described, In particular, for each such J there exists a unique maximal ideal of U containing J and a unique ideal of U minimal with respect to the property that it properly contains JU. Similar results are given in the case when U is the tensor product of infinitely many copies of U(sl ₂).     

Faithful Inner Ideals in Jbw*-Triples

The kernel Ker(J) and the annihilator J^⊥ of a weak*-closed inner ideal J in a JBW*-triple A consist of the sets of elements a in A for which {J a J} and {J a A} are zero, respectively, and J is said to be faithful if, for every non-zero ideal I in A, I ∩ Ker (J) is non-zero. It is shown that every weak*-closed inner ideal J in A has a unique orthogonal decomposition into a faithful weak*-closed inner ideal f(J) and a weak*-closed ideal f (J)^⊥ ∩ J of A. The central structure of f ( J) is investigated and used to show that J has zero annihilator if and only if it coincides with the multiplier of f (J). The results are applied to the cases in which J is the Peirce-two or Peirce-zero space A₂(v) or A₀(v) corresponding to a tripotent v in A, and to the case in which the JBW*-triple A is a von Neumann algebra.     

Bertini theorems for ideals linked to a given ideal

We prove a generalization of Flenner’s local Bertini theorem for complete intersections. More generally, we study properties of the ‘general’ ideal linked to a given ideal. Our results imply the following. LetR be a regular local Nagata ring containing an infinite perfect fieldk, andI?R is an equidimensional radical ideal of heightr, such thatR/I is Cohen-Macaulay and a local complete intersection in codimension 1. Then for the ‘general’ linked idealJ _α, R/J_α is normal and Cohen-Macaulay. The proofs involve a combination of the method of basic elements, applied to suitable blow ups.     

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec _r (R) (resp. Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U _r (eR) = {P ? Spec _r (R)| e ? P}. Let  = ∪_{P?Prim r (R)} Spec _r ^P (R), where Spec _r ^P (R) = {Q ?Spec _r ^P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on  and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if  is a space with Zariski topology if and only if, for any Q ? , Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ? = Max _r (R) ∪  Prim _r (R) (resp.  = Prim _r (R)) there is an element e in R with e ² ? e ? J(R) such that U = U _r (eR) ∩  ? (resp. U = U _r (eR) ∩  ), where J(R) is the Jacobson of R. (3) Max _r (R) ∪  Prim _r (R) is connected if and only if Max _l (R) ∪  Prim _l (R) is connected if and only if Prim _r (R) is connected.     

THE M-VALUATION SPECTRUM OF A COMMUTATIVE RING

Abstract

In 1945, N. Jacobson has introduced the definition of radical of a ring A (which is known as “Jacobson radical”, and is denoted J = J(A)). Later the concept of (Jacobson) radical of a left (or right) A-module M, J(M), has been defined as the intersection of all submodules N ≤ M such that M/N is simple. Thus one may consider the left radical J _l  = J( _A A) and the right radical J _r  = J(A _A ) of A, which are bilateral ideals of A, and are contained in J(A). If A has identity, one has J = J _l  = J _r , but this equality is not valid in general. Dual, it is possible to define left socle S _l and right socle S _r of A. We shall establish relations between J, J _l , J _r , S _l and S _r , and for artinian algebras we shall obtain expressions for J _l (A) and J _r (A), S _l (A) and S _r (A). In particular, if A is a finite dimensional algebra over a field we display J _l  = J( _A A) (and J _r  ? J(A _A )) in a matrix representation.     

Gelfand Factor Rings and Weak Zariski Topologies

Let R be any ring with identity. Let N(R) (resp. J(R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec _r (R) (resp. Spec _l (R), Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec _r (R) (with weak Zariski topology). The following results are obtained: (1) R/N(R) is a Gelfand ring if and only if Spec _r (R) is a normal space if and only if Spec _l (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.     

Relative K0, Annihilators, Fitting Ideals and Stickelberger Phenomena

When G is abelian and l is a prime we show how elements of therelative K-group K₀(Z_l[G], Q_l give rise to annihilator/Fittingideal relations of certain associated Z[G]-modules. Examplesof this phenomenon are ubiquitous. Particularly, we give examplesin which G is the Galois group of an extension of global fieldsand the resulting annihilator/Fitting ideal relation is closelyconnected to Stickelberger's Theorem and to the conjecturesof Coates and Sinnott, and Brumer. Higher Stickelberger idealsare defined in terms of special values of L-functions; whenthese vanish we show how to define fractional ideals, generalisingthe Stickelberger ideals, with similar annihilator properties.The fractional ideal is constructed from the Borel regulatorand the leading term in the Taylor series for the L-function.En route, our methods yield new proofs, in the case of abeliannumber fields, of formulae predicted by Lichtenbaum for theorders of K-groups and étale cohomology groups of ringsof algebraic integers. 2000 Mathematics Subject Classification11G55, 11R34, 11R42, 19F27.     

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.     

Job shop scheduling with unit time operations under resource constraints and release dates

We consider the problem of job shop scheduling with m machines and n jobs J_i, each consisting of l_i unit time operations. There are s distinct resources R_h and a quantity q_h available of each one. The execution of the j-th operation of J_i requires the presence of u_ijh units of R_h, 1 ≤i≤n, 1 ≤j≤l_i, and 1 ≤h≤s. In addition, each J_i has a release date r_i, that is J_i cannot start before time r_i. We describe algorithms for finding schedules having minimum length or sum of completion times of the jobs. Let l=max{l_i} and u=|{u_ijh}|. If m, u and l are fixed, then both algorithms terminate within polynomial time.