关于拟平坦模的几个注记

本文所有的环均指有单位元的环,模均指酉模。左R-模M称为拟内射的,如果对任意N相似文献   

Two questions on semi-clean group rings

It was proved in [4] that every group ring of a torsion abelian group over a commutative local ring is a semi-clean ring. It was asked in [4] whether every group ring of a torsion abelian group over a commutative clean ring is a semi-clean ring and whether every group ring of a torsion abelian group over a commutative semi-clean ring is a semi-clean ring. In this paper, we give a positive answer to question 1 and a negative answer to question 2.     

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.     

Noether环理想的性质

Noether交换环是一类非常重要的环,本文主要对Noether交换环进行了研究和讨论,得到了Noether交换环、Noether整环的若干性质;并推广了文[1]中的部分结果.     

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.     

RD-Flatness and RD-Injectivity

It is proven that every commutative ring whose RD-injective modules are Σ-RD- injective is the product of a pure semisimple ring and a finite ring. A complete characterization of commutative rings for which each Artinian (respectively simple) module is RD-injective, is given. These rescan be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively the RD-relativization of flat modules and fp-injective modules. It is also shown that a commutative ring is perfect if and only if each RD-flat module is RD-projective.     

半质环的交换性定理

任意半质PI－环的交换性问题是环的交换性理论的一个主要内容,本文利用亚直不可约环和分式环的性质,研究了任意半质PI－环的交换性问题,证明了半质PI－环的交换性定理。     

Torsion-free weakly transitive E-Engel Abelian groups

It is proved that if all the endomorphisms of a reduced torsion-free weakly transitive Abelian group are bounded right-nilpotent, then its ring of endomorphisms is commutative. The ring of endomorphisms of a torsion-free Abelian group with periodic group of automorphisms and Engel ring of endomorphisms is also commutative.     

The ring of quotients of R(S)

A ring of quotients of the semigroup ring R(S) is discussed where R has a σ-set Σ and S has a σ-set Δ. In particular, we study the cases where (1) R is an integral domain and S is a commutative cancellative semigroup, (2) R is a commutative ring and S is a semilattice and (3) R is a commutative ring and S is a Rees matrix semigroup over a semigroup. Communicated by G. Lallement     

*-Orderings on a ring with involution

The object of the paper is to extend part of the theory of *-orderings on a skewfield with involution to a general ring with involution. The valuation associated to a *-ordering is examined. Every *-ordering is shown to extend. *-orderings are shown to form a space of signs as defined by Brocker and Marshall. In case the involution is the identity, the ring under consideration is commutative and the *-orderings are just the usual orderings making up the usual real spectrum of a commutative ring as defined by Coste and Roy.     

Isomorphism of generalized matrix rings

An isomorphism problem is considered for generalized matrix rings with values in a given ring R. An exhaustive answer is given for the case of a commutative domain R and a commutative local ring R.     

Weakly Clean and Exchange Uni Rings

Ukrainian Mathematical Journal - We find a criterion for a commutative ring to be a weakly clean UNI ring, as well as a criterion for an exchange ring to have strongly invo-clean unit group. These...     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

保持交换环上两类矩阵运算的映射

主要针对交换环上两类矩阵的保持问题进行展开:(1)刻画了交换环上全矩阵空间和上三角形矩阵空间的保持反对合矩阵映射的形式.(2)研究了交换环上n阶上三角形矩阵空间的保持伴随矩阵映射的形式.     

Decomposition of ideals into pseudo-irreducible ideals

A proper ideal of a commutative ring is called pseudo-irreducible if it cannot be written as a product of two comaximal proper ideals. In this paper, we give a necessary and su?cient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal pseudo-irreducible ideals. Examples of such rings include Laskerian rings, or more generally J-Noetherian rings and ZD-rings. We study when certain classes of rings satisfy this condition.     

交换L*-环

我们证明一个交换整环或交换局部环是一个L*-环当且仅当它是一个O*-环.文中还证明了一些一般性的条件使之一个交换环不是L*.     

On α-Relation and Transitivity Conditions of α

The main tools in the theory of hyperstructurs are the fundamental relations. The fundamental relation on hyperring was introduced by Vougiouklis at the fourth AHA congress (1990). The fundamental relation on a hyperring is defined as the smallest equivalence relation so that the quotient would be the ring. Note that, generally, the commutativity in the ring are not assumed. In this article, we introduce a new strongly regular equivalence relation on hyperring so that the quotient is a commutative ring. Also we state the condition that is equivalent with the transitivity of this relation and finally we characterize the complete hyperring (with the fundamental relation as commutative).     

Commutative ideal theory without finiteness conditions: Completely irreducible ideals

An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those commutative rings in which every ideal is an irredundant intersection of completely irreducible ideals.

     

The structure of commutative rings with monomorphic flat envelopes

It is obvious that OF and Von Neumann regular rings have monomorphic flat envelopes. In this paper we completely describe the structure,in terms of OF and Von Neumann regular rings, of those commutative rings all of whose modules have a monomorphic flat envelope (m.f.e. ). For that, we introduce the notion of locally QF ring with m.f.e., whose structure is given in terms of OF rings. It turns out that a commutative ring R with m.f.e. is characterized as a (essential) subdirect product of a locally QF ring with m.f.e. and a Von Neumann regular ring, with the latter flat as an R-module.     

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R.