Linear ultrafilters

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are “continuous” with respect to U modulo the ideal of those that are “trivial” with respect to U forms a division ring E(U). (These division rings can also be described as the endomorphism rings of the simple left End(X)-modules.) We study this and the dual construction, based on maximal cofiIters of subspaces of X, in particular, the relation between the constructed division rings and the original field or division ring k. We end by examining a more general construction in which X is a module over a general ring, given with both a filter and a cofilter of submodules.     

Approximately Cohen-Macaulay Rings and Samuel Functions

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Rings Related to Mininjective and Min-Flat Modules

R is called a left PS (resp. left min-coherent, left universally mininjective) ring if every simple left ideal is projective (resp. finitely presented, a direct summand of R). We first investigate when the endomorphism ring of a module is a PS ring, a min-coherent ring, or a universally mininjective ring. Then we characterize PS rings and universally mininjective rings in terms of endomorphisms of mininjective and min-flat modules. Finally, we study commutative min-coherent rings and (universally) mininjective rings using properties of homomorphism modules of special modules.     

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     

广义幂级数环的本质理想和非奇异性

本文研究了广义幂级数环与其系数环在本质理想和非奇异性上的关系.利用本质理想的定义和性质,得到了广义幂级数环的左理想为本质左理想的菪干充分必要条件.在此基础上,给出了广义幂级数环为左非奇异环的充分必要条件.     

Sur les algebres de Rees II

The notion of a Rees ring was introduced in view of what one calls today the Artin — Rees lemma. In fact, it is the Rees algebra of an ideal of a commutative ring with identity. We give in this paper a number of results which concern the Rees algebra of a module over a commutative ring with identity which also complete those of a previous paper (cf. [7]). In particular, we show that the Rees algebra of a module can be approached, in a sense which is made precise in the paper, through tensor or symmetric flat algebras.     

Strongly rigid and I-rigid rings

In this paper we consider questions connected with the problem of rigidity of rings for the class of finite-rank torsion-free rings (A ring is called rigid, if it has only trivial endomorphisms). We study strongly rigid and I-rigid rings [3] (A ring R is called strongly rigid, if any ring quasi-isomorphic to R is rigid). The main results of this paper are the characterization of strongly rigid rings and the establishment of relations between strongly rigid and I-rigid rings. It is shown that the verification of strong rigidity of the ring reduces to testing the rigidity of some field of algebraic numbers over its subfield. This gives rise to examples of non-trivial strongly rigid rings.     

Universal deformation rings and Klein four defect groups

In this paper, the universal deformation rings of certain modular representations of a finite group are determined. The representations under consideration are those which are associated to blocks with Klein four defect groups and whose stable endomorphisms are given by scalars. It turns out that these universal deformation rings are always subquotient rings of the group ring of a Klein four group over the ring of Witt vectors.

     

Rationality for generic toric rings

We study generic toric rings. We prove that they are Golod rings, so the Poincaré series of the residue field is rational. We classify when such a ring is Koszul, and compute its rate. Also resolutions related to the initial ideal of the toric ideal with respect to reverse lexicographic order are described. Received August 13, 1997; in final form October 23, 1998     

Projektive Moduln über Polynomringen A[T1,...,Tm] mit einem regulären Grundring A

We are concerned with a question of H. Bass and D. Quillen which asks whether the following is true:If A is a regular noetherian ring, then every finitely generated projective module P over a polynomial extension A[T] of A is extended from A. (cf. [1], § 4, (IX) and [7]).We give an affirmative answer, either if (i) A equals a ring of fractions of a polynomial ring over a regular noetherian ring B with dim B2, or if (ii) A equals a ring of fractions of a polynomial extension of a power series ring over a complete regular local ring B with dim B2.(ii) implies the case that A is an unramified complete regular local ring. This generalizes the result in [4], which has been proved independently in [5]. (i) spezializes to the known theorem of Quillen and Suslin if A=B (cf. [2]).     

On the endomorphism ring of a module noetherian with respect to a torsion theory

IfM is a module torsionfree and noetherian with respect to a torsion theory, ifS is the endomorphism ring ofM, and ifL(S) is the ideal ofS consisting of all endomorphisms with large kernels, thenL(S) is nilpotent and a bound on the index of nilpotency ofL(S) is given.     

Semiprime rings with bounded indices of nilpotent elements

This paper deals with the structure of semiprime rings for which the indices of the nilpotent elements are bounded. It is shown that the complete right ring of quotients of such a ring is a regular, right self-injective ring in which each finitely generated ideal is generated by a central idempotent. The indices of the nilpotent elements of the factor ring of such a ring with respect to a minimal prime ideal do not exceed the upper bound of the indices of the nilpotent elements of the original ring. A criterion for the regularity (in the sense of von Neumann) of such rings is obtained. Also investigated are right completely idempotent rings with bounded indices of nilpotent elements (it is shown, in particular, that each nonzero ideal of such a ring contains a nonzero central idempotent).Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 237–249, 1988.     

Semiprimeness,quasi-Baerness and prime radical of skew generalized power series rings

Let R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.     

Lexicodes over finite principal ideal rings

Let R be a (possibly noncommutative) finite principal ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module \(R^n\) is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combinations with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal ideal rings and show that the total ordering of the ring elements has to respect containment of ideals for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.     

A combinatorial principle and endomorphism rings I: Onp-groups

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λ^ℵ ₀) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λ^ℵ ₀≧|R|. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.     

Lifting endomorphisms of formal A-modules over finite fields

Lubin conjectures that for an invertible series to commute with a noninvertible series with only simple roots of iterates, two such commuting power series must be endomorphisms of a single formal group. In this paper, we show that if the reduction of these two commuting power series are endomorphisms of a formal group, then themselves are endomorphisms of a formal group.     

Counter-examples to non-noetherian Elkik's approximation theorem

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.     

SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.     

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.     

Baer and Quasi-Baer Properties of Skew Generalized Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.