关于环的交换性定理

令R为有单位元的结合环,M(R)=N(R)∪J/(R).证明了如果存在正整数m使得所有x,y∈_R\M(R)均满足(xy)~k=x^ky^k(其中k=m,m+1,m+2);或者使得所有x,y∈R\M(R)均满足(xy)~k=y^kx^k(其中k=m-1,m,m+1为正整数),那么R是交换环.     

A generalization of anti-commutative rings arising from 2-varieties

Let R be a non-associative ring of characteristic not 2 or 3 which satisfies the identities (ab+ba)c = (ac+ca)b, a(bc+cb) = b(ac+ca), and a².a = a.a². It is proved that R is power asso-ciative, that if R is simple, then R is either anti-commutative or else commutative and associative. It is shown that if R is nil and semiprime, then R is anti-commutative, and an example of a prime ring of this type which is neither commutative nor anti-commu-tative is given.     

Exchange rings,units and idempotents

An associative ring R with identity is semiperfect if and only if every element of R is a sum of a unit and an idempotent, and R contains no infinite set of orthogonal idempotents. A ring which contains no infinite set of orthogonal idempotents is an exchange ring if and only if every element is a sum of a unit and an idempo-tent     

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’。     

On the Generalized Strongly Nil-Clean Property of Matrix Rings

Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n × n matrix A over R is expressible as a sum A =E1 +…+ Es + N of (commuting) idempotent matrices Ei and a nilpotent matrix N.     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

S-内射模及S-内射包络

设R是环.设S是一个左R-模簇,E是左R-模.若对任何N∈S,有Ext_R~1(N,E)=0,则E称为S-内射模.本文证明了若S是Baer模簇,则关于S-内射模的Baer准则成立;若S是完备模簇,则每个模有S-内射包络;若对任何单模N,Ext_R~1(N,E)=0,则E称为极大性内射模;若R是交换环,且对任何挠模N,Ext_R~1(N,E)=0,则E称为正则性内射模.作为应用,证明了每个模有极大性内射包络.也证明了交换环R是SM环当且仅当T/R的正则性内射包e(T/R)是∑-正则性内射模,其中T=T(R)表示R的完全分式环,当且仅当每一GV-无挠的正则性内射模是∑-正则性内射模.     

关于遗传R-extending模

设R是有单位元的结合环,M是右R－模.本文证明了若M是遗传的R－extending模,则M是Noether一致模的宜和．     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

Commutative semigroups having greatest regular images

Let S be a commutative semigroup. S has a greatest regular image if and only if each of its Archimedean components contains an idempotent. S has a greatest group-with-zero image if and only if S has precisely two Archimedean components and the upper component contains an idempotent. The existence and structure of these images and of greatest group images is related to tensor products.     

Clean Semiprime f-Rings with Bounded Inversion

Abstract

An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently,Anderson and Camillo (Anderson,D. D.,Camillo,V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover,every clean ring is a pm-ring,that is every prime ideal is contained in a unique maximal ideal. In the same article,the authors give an example of a commutative ring which is a pm-ring yet not clean,e.g.,C(?). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1.     

On chained overrings of pseudo-valuation rings

A prime ideal P of a commutative ring R with identity is called strongly prime if aP and bR are comparable for every a, b in R. If every prime ideal of R is strongly prime, then R is called a pseudo-valuation ring. It is well-known that a (valuation) chained overring of a Prufer domain R is of the form R_P for some prime ideal P of R.In this paper, we show that this statement is valid for a certain class of chained overrings of a pseudo-valuation ring.     

Exchange rings,related comparability and power-substitution

An associative ring R with identity is said to satisfy related comparability provided that for any idempotents e, f ∈ R with e = 1+ ab and f = 1+ba for some a, b ∈R, there exists a central idempotent u ∈R such that ueR is isomorphic to a direct summand of ufR and (1-u)fR is isomorphic to a direct summand of (1 - u)eR In this paper, we investigate the power-cancellation properties of modules over exchange rings satisfying related comparability.     

Weyl型代数的环论性质

设A是代数闭域k上有单位元1的交换结合代数，D是A的交换κ-导子组成的非零k-向量空间，苏育才与赵开明引进Weyl型代数A[D]并且证明了结合代数A[D]是单代数当且仅当A是D-单的且k1[D]在A上的作用为忠实的,通过证明A[D]与smash product A#U(D)同构，我们给出了这一结果的一个纯环论的证明，同时给出了A[D]的一个Ore扩张实现。     

On divided commutative rings

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a nonprincipal divided prime, then P^-1 = { x ? T : xP ? P} is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1. Also, we show that if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided, then it is maximal and R is quasilocal.     

Craig interpolation for semilinear substructural logics

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R‐mingle with unit” logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model‐theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property.     

Commutative neat group rings

A ring R is called clean if every element of it is a sum of an idempotent and a unit. A ring R is neat if every proper homomorphic image of R is clean. When R is a field, then a complete characterization has been obtained for a commutative group ring RG to be neat, but not clean. And if R is not a field, then necessary conditions are obtained for a commutative group ring RG to be neat, but not clean. A counterexample is given to show that these necessary conditions are not sufficient.     

The generalized entropic property for a pair of operations

In this paper a generalized entropic property is defined for a pair of operations. We show that for an idempotent algebra A = (A, f, g) with two ternary operations, if one of f or g is commutative and the pair of operations (f, g) satisfies the generalized entropic property, then (f, g) is entropic. Also, it is proved that every idempotent, commutative algebra A = (A, f, g) with a ternary and a binary operation, satisfying the generalized entropic property, is entropic.