Noetherian hereditary abelian categories satisfying Serre duality

In this paper we classify -finite noetherian hereditary abelian categories over an algebraically closed field satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories.

As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.

     

The Noetherian property in some quadratic algebras

We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations--namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2--and on the Ufnarovskii graph . The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.

     

Convergence of sequences of sets of associated primes

It is a well-known result of M. Brodmann that if is an ideal of a commutative Noetherian ring , then the set of associated primes of the -th power of is constant for all large . This paper is concerned with the following question: given a prime ideal of which is known to be in for all large integers , can one identify a term of the sequence beyond which will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

     

On Standard Ideals

In the paper,we study a class of standard ideals which are more general than the m-primary standared ideals discussed in[2].We will prove an important equality concerning I-weak sequences;thus a generalization of the equality of [2] is established.     

关于完全环与Noether环

（ｉ）环Ｒ是左完全环，当且仅当存在一个基数ｃ，使得任意平坦左Ｒ－模是一个拟投射模和一个ｃ－限制的ＥＳ－模的直和。（ｉｉ）Ｒ是左Ｎｏｅｔｈｅｒ环，当且仅当存在一个基数ｃ，使得任意内射左Ｒ－模的直和是一个（拟）连续模和一个ｃ－限制的ＥＳ－模的直和。     

Absolute stable rank and quadratic forms over noncommutative rings

Given an associative ring A, let asr(A) denote the absolute stable range of A, as defined in [5]. We prove that asr(A) 1 + Kdim A if A is a right Noetherian ring, and that asr(A) 1 + cl-Kdim A if A is an affine PI algebra. Combined with results from [5], this provides a cancellation theorem ('Witt cancellation') for quadratic spaces defined over such a ring A.     

Submonoids of Polycyclic-by-Finite Groups and their Algebras

We describe Noetherian semigroup algebras K[S] of submonoids S of polycyclic-by-finite groups over a field K. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal P of K[S] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of S via a generalised matrix ring structure on K[S]/P. Applications to the classical Krull dimension, prime spectrum, and irreducible K[S]-modules are given.     

Semidistributive and distributively decomposable rings

A module is said to be distributive if the lattice of all its submodules is distributive. A module is called semidistributive if it is a direct sum of distributive modules. Right semidistributive rings, as well as distributively decomposable rings, are investigated. Translated fromMatematicheskie Zemetki, Vol. 65, No. 2, pp. 307–313, February, 1999.     

拟δ-Koszul代数

为了研究分次代数的Ext代数的有限生成性,GREEN和MARCOS首次提出了δ-Koszul代数的概念.笔者利用极小分次投射解对δ-Koszul代数的等价刻画,把δ-Koszul代数推广到非分次情形,研究了诺特半完全代数的δ-Koszul性质,并证明了一些性质可以从分次情形遗传到非分次情形.     

关于G-Matlis自反模的换环定理

设Ｒ和Ｔ是Ｎｏｅｔｈｅｒ完备半局部环，Ｒ→Ｔ是环同态．本文证明了，若Ｔ是有限生成或ＡｒｔｉｎＲ－模，Ｍ为Ｇ－Ｍａｔｌｉｓ自反Ｒ－模，则对所有ｎ≥０，Ｅｘｔ^Ｒ_ｎ（Ｔ，Ｍ），Ｅｘｔ^Ｒ_ｎ（Ｍ，Ｔ），Ｔｏｒ^Ｒ_ｎ（Ｔ，Ｍ）以及Ｔｏｒ^Ｒ_ｎ（Ｍ，Ｔ）均是Ｇ－Ｍａｔｌｉｓ自反Ｔ－模．所得结果推广了Ｒ．Ｂｅｌｓｈｏｆ的结果．