关于*-模的若干性质

本文主要给出＊－模的一些充分必要条件,同时还证明了：如果ｆ：Ｒ→Ｓ是环同态,ＲＰ是Ｐ－投射模,当ＲＰ是＊－模,Ｔｉｌｔｉｎｇ模,拟Ｔｉｌｔｉｎｇ模时,Ｓ　ＲＰ也分别是＊－模,Ｔｉｌｔｉｎｇ模,和拟Ｔｉｌｔｉｎｇ模。     

Unique factorization properties in commutative monoid rings with zero divisors

Semigroup Forum - Several different versions of “factoriality” have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the...     

Constants of derivations in polynomial rings over unique factorization domains

A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any -derivation of , where is a commutative field of characteristic zero, is a polynomial ring in one variable over . In this paper we give an elementary proof of this theorem and show that it remains true if we replace by any unique factorization domain of characteristic zero.

     

Normsets and determination of unique factorization in rings of algebraic integers

The image of the norm map from to (two rings of algebraic integers) is a multiplicative monoid . We present conditions under which is a UFD if and only if has unique factorization into irreducible elements. From this we derive a bound for checking if is a UFD.

     

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.     

Almost perfect commutative rings

Almost perfect commutative rings R are introduced (as an analogue of Bazzoni and Salce's almost perfect domains) for rings with divisors of zero: they are defined as orders in commutative perfect rings such that the factor rings $R/Rr$ are perfect rings (in the sense of Bass) for all non-zero-divisors$r\in R$. It is shown that an almost perfect ring is an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains. Noetherian almost perfect rings are exactly the one-dimensional Cohen–Macaulay rings. Several characterizations of almost perfect domains carry over practically without change to almost perfect rings. Examples of almost perfect rings with zero-divisors are abundant.     

Extensions of commutative rings

In studying the minimal prime spectra of commutative rings with identity we have been able to identify several interesting types of extensions of rings. In particular, we determine what kind of ring extensions will result in a homeomorphisms of the hull-kernel and inverse topologies on the minimal prime spectra. We relate these types of extensions to other known types of extensions.     

Representation of real commutative rings

During the last 10 years there have been several new results on the representation of real polynomials, positive on some semi-algebraic subset of . These results started with a solution of the moment problem by Schmüdgen for compact semi-algebraic sets. Later, Wörmann realized that the same results could be obtained by the so-called “Kadison–Dubois” Representation Theorem.The aim of our paper is to present this representation theorem together with its history, and to discuss its implication to the representation of positive polynomials. Also recent improvements of both topics by T. Jacobi and the author will be included.     

Clean matrices over commutative rings

A matrix A ∈ M _n (R) is e-clean provided there exists an idempotent E ∈ M _n (R) such that A-E ∈ GL _n (R) and det E = e. We get a general criterion of e-cleanness for the matrix [[a ₁, a ₂,..., a _n +1]]. Under the n-stable range ondition, it is shown that [[a ₁, a ₂,..., a _n +1]] is 0-clean iff (a ₁, a ₂,..., a _n +1) = 1. As an application, we prove that the 0-cleanness and unit-regularity for such n × n matrix over a Dedekind domain coincide for all n ⩾ 3. The analogous for (s, 2) property is also obtained.      

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].     

Commutative rings with two-absorbing factorization

We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local TAF-domains are the atomic pseudo-valuation domains.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Polynomial orbits in finite commutative rings

Let R be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.     

Associated primes in commutative polynomial rings

The aim of this paper is to give a new and direct proof of the theorem.