偏序集的关联环的同构问题

设Ｘ是局部有限偏序集（或拟序集），Ｒ是含１的结合环，Ⅰ（Ｘ，Ｒ）是Ｒ上Ｘ的关联环，关联环的同构问题是指：问题１：怎样的环，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）推出偏序集之间的同构Ｘ芒Ｘ’？问题２：怎样的环或偏序集，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｓ）推出Ｒ　Ｓ？本文证明了对唯一幂等元环（非交换），问题１有正面回答；对问题２，我们证明了对交换不可分解环Ｒ、Ｓ，由环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）可得到Ｒ＝Ｓ，Ｘ＝Ｘ’。     

Spectral partially ordered sets

We examine some topics related to (gold)spectral partially ordered sets, i.e., those that are order isomorphic to (Goldman) prime spectra of commutative rings. Among other results, we characterize the partially ordered sets that are isomorphic to prime spectra of rings satisfying the descending chain condition on radical ideals, and we show that a dual of a tree is isomorphic to the Goldman prime spectrum of a ring if and only if every chain has an upper bound. We also give some new methods for constructing (gold)spectral partially ordered sets.     

The probability of generating a classical group

We show that a countable totally and discretely ordered set with first element inherently carries the structure of an ordered commutative euclidean monoid, provided its order type is of a certain kind. As an application we specify the order types of all discretely ordered sets which can be expanded to ordered commutative euclidean monoids.     

Automorphisms of the semigroup of nonnegative invertible matrices of order two over partially ordered commutative rings

In the paper, the automorphisms of the semigroup of nonnegative invertible matrices of order two over a partially ordered commutative ring with 2 invertible are described.     

The ordering of commutative terms

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a, b write a ⩽ b if b contains a subterm that is a substitution instance of a. With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has no automorphisms except the identity.     

Differential Calculi on Quantum Groupoids

Let R be a commutative ring with Noetherian spectrum in which zero is a primary ideal. We determine the minimal zero-dimensional extensions of R when every regular prime ideal of R is contained in only finitely many prime ideals. This extends previous results of the first author for dim (R) ≤1. We also present a characterization of the partially ordered set of prime ideals in a ring with Noetherian spectrum.     

Representations of finite partially ordered sets over commutative artinian uniserial rings

Categories of representations of finite partially ordered sets over commutative artinian uniserial rings arise naturally from categories of lattices over orders and abelian groups. By a series of functorial reductions and a combinatorial analysis, the representation type of a category of representations of a finite partially ordered set S over a commutative artinian uniserial ring R is characterized in terms of S and the index of nilpotency of the Jacobson radical of R. These reductions induce isomorphisms of Auslander-Reiten quivers and preserve and reflect Auslander-Reiten sequences. Included, as an application, is the completion of a partial characterization of representation type of a category of representations arising from pairs of finite rank completely decomposable abelian groups.     

Totally ordered sets and the prime spectra of rings

Let T be a totally ordered set and let D(T) denotes the set of all cuts of T. We prove the existence of a discrete valuation domain O_v such that T is order isomorphic to two special subsets of Spec(O_v). We prove that if A is a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set U?Spec(A) such that the prime spectrum of A is order isomorphic to D(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view.     

Noncommutative Topologies,Localization, and Sheaves

In this note, we introduce and study the notion of sheaf on a noncommutative topology and construct an associated sheafification functor in this noncommutative context. It appears that abstract localization in the category of presheaves on a partially ordered set provides an elegant way to define this sheafification functor, generalizing similar constructions in the commutative case.     

Local derivations of finitary incidence algebras

Let P be a partially ordered set, R a commutative ring with identity and FI(P,R) the finitary incidence algebra of P over R. We prove that each R-linear local derivation of FI(P,R) is a derivation, which partially generalizes Theorem 3 of [21].     

可换序半群的分式扩张

In this paper we extend the construction of the field of rational numbers from the ring of integers to an arbitrary commutative ordered semigroup.We first construct a fractional ordered semigroup and a homomorphism ψs:R→S-1R.Secondly,we characterize the commutative ordered semigroup so constructed by a universal mapping property.     

Shellability of Interval Orders

An finite interval order is a partially ordered set whose elements are in correspondence with a finite set of intervals in the line, with disjoint intervals being ordered by their relative position. We show that any such order is shellable in the sense that its (not necessarily pure) order complex is shellable.     

Integral Stone Rings

It is shown that certain partially ordered rings, defined by some of the properties of the totally ordered ring of integers, are exactly the bounded Z-rings, that is, the commutative f-rings with strong singular unit. The partially ordered rings in question amount to a discrete version of the rings introduced by M.H. Stone for his abstract characterization of the rings of real-valued continuous functions on compact Hausdorff spaces, and the function rings they correspond to are given by the integer-valued continuous functions on Boolean spaces.     

Lifting standard bases in filtered structures

In this paper we further study the theory of standard bases in filtered structures as introduced by Robbiano. In particular, we study the situation in which a commutative ring A is filtered by commutative ordered monoids △ and τ in such a way that the τ filtration refines the △ filtration. We use these results to determine conditions under which a A standard basis can be lifted to a τ standard basis. Finally, we interpret these results in three situations where the refinements are given by changing the status of the variables in a polynomial ring, refining a partial order on the monomials in a polynomial ring, and refining a filtration by an I-adic filtration for an ideal I in the ring A.     

Ordered *-Algebras with an Order Unit

We develop a theory of ordered *-algebras with an order unit. These are complex algebras equipped with a conjugate-linear vector space involution whose hermitian elements form an ordered real algebra with an order unit but not necessarily with a multiplicative identity. Our main result is a representation theorem for ordered *-algebras with an order unit. This can be viewed as the complex and the non-unital version of the representation theorem of Stone and Kadison for ordered real algebras containing an order unit which is a multiplicative identity. A key role in our approach is played by certain weighted function algebras.     

有单位元交换环上二阶矩阵的因子分解

对有单位元交换环上矩阵分解问题进行了讨论,给出了有单位元交换环上二阶矩阵可以因式分解的充分必要条件,即单位元交换环上二阶矩阵可以因式分解当且仅当这个矩阵的行列式可以因子分解.     

Quillen-Suslin theory revisited

We partially answer a question of T.Y. Lam by obtaining partial generalized Local Global and Monic Inversion Principles for the action of the elementary subgroup on the set of unimodular rows over a commutative ring.     

Free ordered modules

We establish the necessary and sufficient condition on a partially ordered set S such that a free ordered R-module (R is a linearly ordered ring without divisors of zero) over the set S is o-isomorphic with a free ordered R-module over a trivially ordered set.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 477–487, October, 1972.     

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.