Noetherian-like properties in polynomial and power series rings

There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions $R[X_1??[X_n?$ over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that $R/P$ is integrally closed for each prime ideal P of R, then $R[X]$ is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then $R[X_1,?,X_n]$ is an SFT-ring. Finally, we show that the power series extension $R?X?$ over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian.     

Classification of relation types of Ore extensions of dimension 5

To study AS-regular algebras of dimension 5, we consider dimension 5 graded iterated Ore extensions generated in degree one. We classify the possible degrees of relations and structure of the free resolution for extensions with 3 and 4 generators. We show that every known type of algebra of dimension 5 can be realized by an Ore extension and we consider which of these types cannot be realized by an enveloping algebra.     

On right G-semilocal rings

The notion of semilocal ring is extended to the classes of right G-semilocal and right N-semilocal rings. We explore the algebraic properties of such classes and study their relations with many other rings such as clean, exchange and I-finite rings. Localization of right G-semilocal rings is considered.     

Criteria for -ampleness

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a -ample divisor, where is an automorphism of a projective scheme . Many open questions regarding -ample divisors have remained.

We derive a relatively simple necessary and sufficient condition for a divisor on to be -ample. As a consequence, we show right and left -ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms yield a -ample divisor.

     

Noether环上加性秩函数对应的Gabriel拓扑与相应商环的构造

本文利用［１］中建立的关系,给出了半素右Ｎｏｅｔｈｃｒ环上所有加性秩函数对应的Ｇａｂｒｉｅｌ拓朴与相应商环的构造,并建立了一般Ｎｏｅｔｈｅｒ环上加性秩函数与Ａｒｔｉｎ环上模范畴中的有限长之间的联系．     

Serial Rings with T-Nilpotent Prime Radical

In this paper we consider serial rings with T-nilpotent prime radical, factor-rings of which by the prime radical are right Noetherian rings. We prove that the prime quiver of such a ring is a disconnected union of cycles and chains. In the case when the prime quiver of such a serial ring is a chain the prime radical is nilpotent. For serial rings with nilpotent prime radical we introduce an analogue of Kupisch series. Presented by Yu. Drozd Mathematics Subject Classifications (2000) 16P40, 16G10.     

Envelopes of commutative rings

Given a significative class ℱ of commutative rings, we study the precise conditions under which a commutative ring R has an ℱ-envelope. A full answer is obtained when ℱ is the class of fields, semisimple commutative rings or integral domains. When ℱ is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.     

Semigroups presented by one relation and satisfying the Church-Rosser property

This paper is devoted to the study of semigroups presented by a single defining relationA=B and satisfying the Church-Rosser property. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 114–118, January, 1997. Translated by A. I. Shtern     

On Semiperfect Semidistributive Rings

Mathematical Notes -