环的（λ,μ）-模糊同态

模糊同态是模糊代数学的重要概念之一,它可由不同的模糊映射产生。本文利用（λ,μ）-模糊映射给出了环的（λ,μ）-模糊同态的定义,从而研究了（λ,μ）-模糊同态下（λ,μ）-模糊子环和（λ,μ）-模糊理想的对应关系及若干性质,最后建立了环的（λ,μ）-模糊同态基本定理。     

Automorphisms of a chain ring

Let A be a chain ring that is a faithful algebra over a commutative chain ring R, such that is a separable, normal, algebraic field extension of and is countably generated over . It has been recently proved by Alkhamees and Singh that A has a coefficient ring R ₀, and there exists a pair (θ, σ) with θ ∈ A, σ an R-automorphism of R ₀ such that J (A) = θ A = Aθ, and θa = σ (a) θ, a ∈ R ₀. The question of the extension of certain R-automorphisms of R ₀ to R-automorphisms of A is investigated. Mathematics Subject Classification (2000) Primary 16H05, 16W20, secondary 13F20, 13J15     

Categorical Realizations of Quivers

We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers (in the sense of [9 Gothen, P. B., King, Alastair D. (2005). Homological algebra of twisted quiver bundles. J. London Math. Soc. (2) 71(1):85–99.[Crossref], [Web of Science ®] , [Google Scholar]]), and bilinear pairings as special cases. In this general context, we prove the cancelation from direct sums, and show the existence and uniqueness of decomposition into a sum of indecomposable objects, provided certain assumptions hold. This yields similar results for the special cases just mentioned. Using similar ideas, we also prove a version of Fitting's Lemma for natural transformations between functors.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants

Let F be an affine flat group scheme over a commutative ring R, and S an F-algebra (an R-algebra on which F acts). We define an equivariant analogue Q _F (S) of the total ring of fractions Q(S) of S. It is the largest F-algebra T such that S ? T ? Q(S), and S is an F-subalgebra of T. We study some basic properties.

Utilizing this machinery, we give some new criteria for factoriality (unique factorization domain property) of (semi-)invariant subrings under the action of affine algebraic groups, generalizing a result of Popov. We also prove some variations of classical results on factoriality of (semi-)invariant subrings. Some results over an algebraically closed base field are generalized to those over an arbitrary base field.     

单大环与单小环的结构

本文给出了单大环与单小环的结构。     

On the lattice of matric-extensible radicals. II

Summary We continue our study of the lattice of matric-extensible radicals of associative rings. We find some atoms generated by simple rings of the lattices of all matric-extensible radicals, matric-extensible supernilpotent radicals and matric-extensible special radicals. We consider *-rings, which were previously defined by the second author, and consider when they generate atoms of these lattices.     

The Jacobson Radical of （R[G]）^G

In this paper,the Jacobson radical of (R[G])G, the fixed point subring of a group ring (R[G]) is studied. Some related properties of (R[G])G are discussed.     

Maximal and minimal prime ideals of incidence algebras

Abstract

In this paper we study the relationship between some homological properties such as the weak and the global dimension, the strong n-coherence, and the (n, d)-property, where n and d are two integers, of a commutative ring and its subrings retract. A special application is the transfer of these properties from a commutative ring to its fixed subring with respect to a subgroup of its group of automorphisms. It concludes with a discussion of the scopes and limits of our results.