Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

Commutative Local Rings Whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in Behboodi et al. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules (J. Algebra 345:257–265, 2011) the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.     

Lazy bases: a minimalist constructive theory of Noetherian rings

We give a constructive treatment of the theory of Noetherian rings. We avoid the usual restriction to coherent rings; we can even deal with non‐discrete rings. We introduce the concept of rings with certifiable equality which covers discrete rings and much more. A ring R with certifiable equality can be fitted with a partial ideal membership test for ideals of R. Lazy bases of ideals of R [X ] are introduced in order to derive a partial ideal membership test for ideals of R [X ]. It is then proved that if R is Noetherian, then so is R [X ]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local Duality for Modules over Noetherian Commutative Rings

Some applications of the general theorem on the existence of local duality for modules over Noetherian commutative rings are given. Let be a Noetherian commutative ring, let be a set of maximal ideals in , and let . Then the category of Artinian modules is dual to the category of Noetherian modules. Several structural results are proved, including the theorem on the structure of Artinian modules over principal ideal domains. For rings of special kinds, double centralizer theorems are proved. Bibliography: 5 titles.     

Structure of rings of functions with Riemann Stieltjes convolution products

In this note three sets of complex valued functions with pointwise addition and a Riemann Stieltjes convolution product are considered. The functions considered are discrete analytic functions, sequences, and continuous functions of bounded variation defined on the nonnegative real numbers. Each forms a commutative algebra with identity. The discrete analytic functions form a principal ideal ring with five maximal ideals, nine prime ideals, and is essentially a direct sum of four discrete valuation rings. The ring of sequences is isomorphic to an ideal of the ring of discrete analytic functions; it has two maximal and three prime ideals. Both contain divisors of zero. The units, associates, irreducible elements and primes in these two rings are described. The results are used to study the continuous functions; partial results are obtained concerning units and divisors of zero. The product satisfies a convolution theorem.     

Semiperfect Prüfer rings and FPF rings

The Asano-Michler theorem states that a 2-sided order R in a simple Artinian ringO is hereditary provided thatR satisfies the three requirements: (AM1) Noetherian; (AM2) nonzero ideals are invertible; (AM3) bounded. We generalize this in one direction by specializing to a semiperfect bounded orderR, and prove thatR is semihereditary assuming only that finitely generated nonzero ideals are invertible (=R is Prüfer). In this case,R ≈ a fulln ×n matrix ringD _n over a valuation domainD. More generally, we study a ringR, called right FPF, over which finitely generated faithful right modules generate the category mod-R of all rightR-modules. We completely determine all semiperfect Noetherian FPF rings: they are finite products of semiperfect Dedekind prime rings and Quasi-Frobenius rings. (For semiprime right FPF rings, we do not require the Noetherian or semiperfect hypothesis in order to obtain a decom-position into prime rings: the acc on direct summands suffices. The “theorem” with “semiperfect” delected is an open problem.     

Quantale中的素根定理

在Quantale中引入了m系的概念,利用m系讨论了Quantale中素理想和半素理想之间的关系.在此基础上证明了当Q是可换Quantale时,Id(Q)是空间式Quantale当且仅当Q中的任一理想都是半素理想.最后把环和序半群中的素根定理推广到Quantak中,得到了Quantale中的素根定理.     

Über B. Buchbergers verfahren,systeme algebraischer gleichungen zu lösen

An algorithm of B. Buchberger's is extended to polynomial rings over a Noetherian ring. In a specialized version, it can be used for computing “elimination ideals”. Over fields, it provides the determination of the minimal prime ideals which contain the given ideal, except that the primeness must be proved with other methods. Estimates for computing time are not given.     

Generalised Chain Conditions,Prime Ideals,and Classes of Locally Finite Lie Algebras

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radicalof any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is qussi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.     

Artin环与Noether环的一个刻画

在[1]文中利用极大左理想刻画了Noether环,本文引进Noether左理想、Artin左理想、m左理想等概念(当I是环R的极大左理想时, I既是Noether、Artin的也是m的,此时m=1。),证明了[1]文中相应的结论,给出了相应的Artin环的刻画。 定义1 环R的左理想I称为Artin(Noether),如果R/I是Artin(Noether)R模。 定义2 环R的左理想I称为m理想,如果R/I的任何R子模都可由m个元生成。 本文的主要结论:     

Polynomes a valeurs entieres sur un anneau de pseudovaluation

We study the ring of integral valued polynomials over a pseudovaluation domain A. We entirely determine the set of prime ideals above the maximal ideal M of A: if M is a principal ideal in the valuation domain V associated with A and if its residue field is finite, then this set is in bijection with a topologically complete ring, as in the Noetherian case; if M is principal but of infinite residue field in V, then this set is finite; at last, if M is not principal, then the ring of integral valued polynomials is included in V[X] and has the same set of prime ideals above M.     

A principal ideal theorem analogue for modules over commutative rings

The Principal Ideal Theorem states that if Re is a commutative Noetherian ring and ? is a prime ideal of Re which is minimal over a principal ideal then Pe has height at most 1. Also, if Re is a (not necessarily Noetherian) UFD and Pe is a prime ideal of Re minimal over a principal ideal then Pe has height at most 1. We shall show that there are analogues for modules over commutative rings, but they hold only in special cases.     

Lattice Modules over Principal Element Domains

Abstract

This paper obtains a structure theorem for a finitely generated lattice module 𝔐 over a Noetherian principal element domain 𝔏,with a slightly stronger theorem if the lattice module satisfies a hypothesis valid over principal ideal domains. Additionally,we obtain a new characterization of Dedekind lattice domains as multiplicative lattice domains over which there exists a non torsion,principally generated,Noetherian join-principal-element lattice module.     

On noncommutative piecewise Noetherian rings

We extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.     

A noncommutattve analog of the cohen theorem

By using weakly primary right ideals, we prove an analog of the Cohen theorem for rings of principal right ideals.     

Domains whose overrings satisfy accp

An integral domain D satisfies ACC on principal ideals (ACJCP) if there does not exist an infinite strictly ascending chain of principal ideals of D. Any Noetherian domain, in particular any Dedekind domain, satisfies ACCP. In this note we prove the following theorem: Let D be an integral domain. Then the integral closure of D is a Dedekind domain if and only if every overring of D (ring between D and its quotient field) satisfies ACCP.     

关于强正则环的刻画

通过单边理想是广义弱理想来刻画强正则环,证明了下列条件是等价的:①R是强正则环;②R是半素的左GP-V′-环,且每一个极大的左理想是广义弱理想;③R是半素的左GP-V′-环,且每一个极大的右理想是广义弱理想.     

关于半准素环

交换环Ｒ称为（受限制的）半准素环，如果对Ｒ的每个（非零）主理想Ａ，都有A^1/2是Ｒ的素理想，本文刻画了受限制的半准素环，给出了有单位元的Ｎｏｅｔｈｅｒ受限制的半准素环的分类以及半准素整环是伪赋值整环的一个条件     

When does the subadditivity theorem for multiplier ideals hold?

Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.

     

满足(accr)环的I-adic完备化及其平坦性

满足（ａｃｃｒ）环是比Ｎｏｅｔｈｅｒ环更广泛的一类环，本文讨论了满足（ａｃｃｒ）环的Ｉ－ａｄｉｃ完备化及其平坦性问题，把Ｎｏｅｔｈｅｒ环的有关结果推广到满足（ａｃｃｒ）环上，同时也改进了［１］中的结果．