Some Results for Formal Local Cohomology Modules

Let （R, m） be a commutative Noetherian local ring, I an ideal of R and M a finitely generated R-module. Let limnHm^i（M/I^nM）be the ith formal local cohomology module of M with respect to I.In this paper, we discuss some properties of formal local cohomology modules limnHm^i（M/I^nM）,which are analogous to the finiteness and Artinianness of local cohomology modules of a finitely generated module.     

Cofiniteness of Lo cal Cohomology Mo dules with Resp ect to a Pair of Ideals

Let R be a commutative Noetherian ring, I and J be two ideals of R, and M be an R-module. We study the cofiniteness and finiteness of the local cohomology module HiI,J（M） and give some conditions for the finiteness of HomR（R/I, HsI,J（M）） and Ext1R（R/I, HsI,J（M））. Also, we get some results on the attached primes of HdimMI,J （M）.     

Some criteria for the Cohen-Macaulay property and local cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.     

On（N,U）-Coherence of Modules and Rings

Let U be a flat right R-module and N an infinite cardinal number.A left R-module M is said to be (N,U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (N,U)-finitely presented in σ[M].It is proved under some additional conditions that a left R-module M is (N,U)-coherent if and only if Л^Ni∈I U is M-flat as a right R-module if and only if the (N,U)-coherent dimension of M is equal to zero.We also give some characterizations of left (N,U)-coherent dimension of rings and show that the left N-coherent dimension of a ring R is the supremum of (N,U)-coherent dimensions of R for all flat right R-modules U.     

Zip模(英文)

A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.     

Pseudop olarity of Generalized Matrix Rings over a Lo cal Ring

Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings and semiregular rings. In this paper, we investigate pseudopolarity of generalized matrix rings K s(R) over a local ring R. We determine the conditions under which elements of K s(R) are pseudopolar. Assume that R is a local ring. It is shown that A ∈ K s(R) is pseudopolar if and only if A is invertible or A2∈ J(K s(R)) or A is similar to a diagonal matrix[u 00 j], where l u-r j and l j-r u are injective and u ∈ U(R) and j ∈ J(R). Furthermore, several equivalent conditions for K s(R)over a local ring R to be pseudopolar are obtained.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

交换环上上三角矩阵的李三导子

Let T(n,R) be the Lie algebra consisting of all n × n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T(n,R)-bimodule.In this paper,we prove that every Lie triple derivation d : T(n,R) → M is the sum of a Jordan derivation and a central Lie triple derivation.     

可换环上上三角矩阵代数的局部若当导子和局部若当自同构

Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.     

Ass（M／N） and Ann（M／N） on the Noetherian Module M

Let R is a noetherian ring,M is a finitely generated R-module.This paper studies the relationbetween associated prime Ass(M/N)and annihilator Ann(M/N),and has given the necessary andsufficient conditions of Ass(M/N)=Ann(M/N).     

On the minimaxness and coatomicness of local cohomology modules

Czechoslovak Mathematical Journal - Let R be a commutative Noetherian ring, I an ideal of R and M an R-module. We wish to investigate the relation between vanishing, finiteness, Artinianness,...     

A natural map in local cohomology

Let R be a Noetherian ring, \(\mathfrak{a}\) an ideal of R, M an R-module and n a non-negative integer. In this paper we first study the finiteness properties of the kernel and the cokernel of the natural map \(f\colon\operatorname{Ext}^{n}_{R}(R/\mathfrak{a},M)\to \operatorname{Hom}_{R}(R/\mathfrak{a},\mathrm{H}^{n}_{\mathfrak{a}}(M))\), under some conditions on the previous local cohomology modules. Then we get some corollaries about the associated primes and Artinianness of local cohomology modules. Finally we will study the asymptotic behavior of the kernel and the cokernel of the natural map in the graded case.     

On the Degree of Hilbert Polynomials of Derived Functors

Mathematical Notes - Given a d-dimensional Cohen–Macaulay local ring (R,m), let I be an m-primary ideal, and let J be a minimal reduction ideal of I. If M is a maximal Cohen–Macaulay...     

算子代数上(m,n)-Jordan导子的刻画

设R是一个环,M是一个R-双边模,m和n是两个非负整数满足m+n≠0,如果δ是一个从R到M的可加映射满足对任意A∈R,(m+n)δ(A~2)=2mAδ(A)+2nδ(A)A,则称δ是一个(m,n)-Jordan导子.本文证明了,如果R是一个单位环,M是一个单位R-双边模含有一个由R中幂等元代数生成的左(右)分离集,那么,当m,n0且m≠n时,每一个从R到M的(m,n)-Jordan导子恒等于零.还证明了,如果A和B是两个单位环,M是一个忠实的单位(A,B)-双边模(N是一个忠实的单位(B,A)-双边模),m,n0且m≠n,U=[A N M B]是一个|mn(m-n)(m+n)|-无挠的广义矩阵环,那么每一个从U到自身的(m,n)-Jordan导子恒等于零.     

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.     

On the Multiplicity and the Cohen-Macaulayness of Fiber Cones of Graded Modules

Let (A,m) be a Noetherian local ring with maximal ideal m,I an ideal of A,J an m-primary ideal of A,N a finitely generated A-module,S =⊕n≥0 Sn a finitely generated standard graded algebra over A and M =⊕n≥0 Mn a finitely generated graded S-module.We characterize the multiplicity and the Cohen-Macaulayness of the fiber cone FJ(M) =⊕n≥0 Mn/JMn.As an application,we obtain some results on the multiplicity and the Cohen-Macaulayness of the fiber cone ⊕n≥0 InN/JInN.     

Cofiniteness of Local Cohomology Modules with Respect to a Pair of Ideals

Let $R$ be a commutative Noetherian ring, $I$ and $J$ be two ideals of $R$, and $M$ be an $R$-module. We study the cofiniteness and finiteness of the local cohomology module $H^i_{I,J} (M)$ and give some conditions for the finiteness of Hom$_R(R/I, H^s_{ I,J} (M))$ and Ext$^1_R(R/I, H^s_{I,J} (M))$. Also, we get some results on the attached primes of $H^{{\rmdim}M}_{I,J} (M)$.     

关于广义分次局部上同调的tame性的一个结果

设R=＋n∈N0Rn（R=R0[R1]）是分次Noether交换环,（R0,m0）是一个局部环,R＋=＋n∈NRn;设N是一个有限生成Z-分次R-模,这里N、N0、Z分别表示全体正整数、全体非负整数和全体格致所构成的集合．令h=sup{i∈Z｜HR＋^i（N）不是Artin模}．Dibaei和Nazari证明了HR＋^h（N）是tame模．我们将该结果推广到了广义分次局部上同调模的情形．     

Rees algebras with minimal multiplicity

Let (R,m) be a local ring. Let S_M denote the Rees algebra S=R[m^rt] localized at its unique maximal homogeneous ideal M=(m,m^rt). Let TN denote the extended Rees algebra T= R[m^rt, t^-1] localized at its unique maximal homogeneous idea N= (t^?1,m,m^r). Multiplicity formulas are developedfor S_M and T_N. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that S_M and T_N are Cohen-Macaulay with minimal multiplicity