An Upper Bound for the w-Weak Global Dimension of Pullbacks

Let R be a commutative ring with identity and I0 an ideal of R.We introduce and study the c-weak global dimension c-w.gl.dim(R/I0) of the factor ring R/I0.Let T be a w-linked extension of R,and we also introduce the wR-weak global dimension wR-w.gl.dim(T) of T.We show that the ring T with wR-w.gl.dim(T) =0 is exactly a field and the ring T with wR-w.gl.dim(T) ≤ 1 is exactly a PwRMD.As an application,we give an upper bound for the w-weak global dimension of a Cartesian square (RDTF,M).More precisely,if T is w-linked over R,then w-w.gl.dim(R) ≤ max{wR-w.gl.dim(T) + w-fdR T,c-w.gl.dim(D) + w-fdn D}.Furthermore,for a Milnor square (RDTF,M),we obtain w-w.gl.dim(R) ≤ max{wR-w.gl.dim(T) + w-fdR T,w-w.gl.dim(D) + w-fdR D}.     

On chains of prime submodules

In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.     

Uniserial modules: sums and isomorphisms of subquotients

Let R be an associative ring with 1. A left R module M is uniserial i f the lattice L(M) of its submodules is totally ordered under inclusion. We give an example of a uniserial module M with the property of having two submodules 0 < H < K < M such that M is isomorphic to K/H (we call a module M with this property shrinkable). Then we give an example of a uniserial module M isomorphic to all its nonzero quotients M/N, N<M, and with L(M) isomorphic to ω²+1; this solves a problem of Hirano and Mogami [7]. Finally we show that for uniserial modules the property of being shrinkable is connected to the problem of deciding whether a module, which is both a homomorphic image of a finite direct sum of uniserial modules and a submodule of a finite direct sum of uniserial modules, is a finite direct sum of uniserial modules     

Near-Rings and Partitioned Groups

Over a commutative ring R with identity, free modules M with 2 distinguished submodules are studied. The category Rep₂R of such objects M have the obvious morphisms between them, which are homomorphisms between .R-modules preserving the distinguished submodules. The endo-morphisms for each M constitute a subalgebra End_RM of the algebra End_RM and the readability of λ-generated R-algebras A as End_RM is considered for cardinals λ. Despite the fact that 4 is the minimal number of distinguished submodules for realizing any algebra over a field il, we are able to prove a similar result in Rep₂R for many rings R including R = Z and algebras which are cotorsion-free. Several examples illustrate the boarder line of our main result. The main theorem is applied for constructing Butler groups in [11]     

On the global dimension of an algebra

Let algebra R = Λ/P, where Λ is a free algebra over a field w. gl. dim R: = {min n ¦? R-modules X, Y, Tor _n+1 ^R (X, Y)=0}. In order that w. gl. dim R≤2n (w. gl. dim R≤2n+1), it is necessary and sufficient that, for any two ideals of algebra Λ, a left ideal A and a right ideal B, containing ideal P, the following equation holds: $$AP^n \cap P^n B = AP^n B + P^{n + 1} (AP^n B \cap P^{n + 1} = AP^{n + 1} + P^{n + 1} B).$$      

非退化扩散过程的水平集和极集

设X(t)(t∈R )是一个d维非退化扩散过程．本文得到了比原有结果更一般的非退化扩散过程极性的充分条件,证明了对任意u∈Rd,紧集E(0, ∞),有若d=1,则对任意紧集F(?)R, 若d≥2,则对任意紧集E ∈(0, ∞), 其中B(Rd)为Rd上的Borel σ-代数,dim和Dim分别表示Hausdorff维数和Packing 维数．     

Gorenstein平坦模与Frobenius扩张

设R■A是环的Frobenius扩张,其中A是右凝聚环,M是任意左A-模.首先证明了_AM是Gorenstein平坦模当且仅当M作为左R-模也是Gorenstein平坦模.其次,证明了Nakayama和Tsuzuku关于平坦维数沿着Frobenius扩张的传递性定理的"Gorenstein版本":若_AM具有有限Gorenstein平坦维数,则Gfd_A(M)=Gfd_R(M).此外,证明了若R■S是可分Frobenius扩张,则任意A-模(不一定具有有限Gorenstein平坦维数),其Gorenstein平坦维数沿着该环扩张是不变的.     

A generalization of finite dimensionality for modules

Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ? C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = $\text{M}$_R, we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ(M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie.     

Homological stabilization for the orthogonal and symplectic groups

It is proved that for a commutative ring R with identity and without finite residue R, fields, the integral groups of homologies Hp(sp_2n(R)) and Hp(O_2n(R)) for a fixed p do not vary with the growth of n only if n⩾2p+dim X Here dim X is the Krull dim X-dimension of the spectrum of the maximal ideals of the ring R. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 222–228, 1987.     

On the spectrum and the geometry of a spherical space form II

In this paper certain relations between the numerical coefficients of the Poisson formula of Part I and geometrical data of a spherical space form M, dim M = 2m+1 are investigated. The results yield an explicit relation between the spectrum of M and the Poincaré map of certain closed geodesics of M. Furthermore, explicit formulas for the multiplicities of the eigenvalues of the Laplacian of M are derived by means of the Poisson formula. At the end of the paper the information about M is examined which is contained in a finite part of spec(M). A partial answer is given in the Corollaries 3 and 6.     

环的B-稳定正则理想

研究了正则理想是B-稳定的充分和必要条件,并且证明环R的正则理想I是B-稳定的当且仅当对任意的有限生成投射右R-模A,如果A₁和A₂是A的有限生成子模且满足A₁≌A₂,A₁=A₁I以及A₂=A₂I,则存在一个有限生成子模B,使得A=A₁（?）B=A₂（?）B;当且仅当对任意的幂等元e,f∈I,eR≌fR蕴含eR/（eR∩fR）≌fR/（eR∩fR）;当且仅当对任意的a∈1+I,存在一个幂等元e∈I,使得a-e∈∪（R）并且aR∩eR=0.进而构造了相关的例子.     

關於Borsuk的絕對同倫擴充性質的討論

<正> 序言.在同倫論中,常常需要考慮滿足這種性質的拓撲空間X設Y為任意的一個正規空間,B為Y的任何一個非空閉集,任何一個由B×(0,1)+Y×(0)到X的映像都可以扩充為一個由Y×(0,1)到X的映像,我們稱這種性質為絕對同倫扩充性質,具有這種性質的空間以及用AHE表示.Borsuk曾經介紹這樣一個重要的定理:     

Chain conditions in modules with krull dimension

Any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals. This result does not hold more generally for modules. In particular if Ris the first Weyl algebra over a field of characteristic 0 then there are Artinian R-modules which do not satisfy the ascending chain condition on prime submodules. However, if Ris a ring which satisfies a polynomial identity then any R-module with Krull dimension satisfies the ascending chain condition on prime submodules, and, if Ris left Noethe-rian, also the ascending chain condition on semiprime submodules.     

拟内射模中的消去定理

当左拟内射模Ｍ的自同态环Ｅｎｄ_RＭ为一Ｄｅｃｋｉｎｄ有限环时．Ｍ的任何两个相互同构的子模的左相关补子横也同构。     

Control subspaces of minimal dimension and root vectors

We investigate the following characteristic of a linear operator A in a Banach space X: disc {inf(dim R′:R′?R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (AⁿR: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system \(\dot x = Ax + Bu\) with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.     

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim _R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.     

Dimension de l’anneau des polynomes a valeurs entieres

Let A be a commutative domain with quotient field K and A_S the ring of integer-valued polynomials thus A_S={f∈K[X]; f(A)⊂A}; we show that the Krull dimension of A_S is such that dim A_S≥dim A[X]-1 and give examples where dim A_S=dim A[X]-1. Considering chains of primes of A_S above a maximal idealm of finite residue field, we give also examples where the length of such a chain can arbitrarily be prescribed (whereas in A[X] the length of such chains is always 1). To provide such examples we consider a pair of domains A⊂B sharing an ideal I such that A/I is finite; we give sufficient conditients to have A_S⊂B[X] and show that in this case dim A_S=dim B[X]. At last, as a generalisation of Noetherian rings of dimension 1, we consider domains with an ideal I such that A/I is finite and a power Iⁿ of I is contained in a proper principal ideal of A; for such domains we show that every prime of A_S above a primem containing I is maximal.      

Remarks on pseudo-valuation rings

A prime ideal P of a ring A is said to be a strongly prime ideal if aP and bA are comparable for all a,b ε A. We shall say that a ring A is a pseudo-valuation ring (PVR) if each prime ideal of A is a strongly prime ideal. We show that if A is a PVR with maximal ideal M, then every overring of A is a PVR if and only if M is a maximal ideal of every overring of M that does not contain the reciprocal’of any element of M.We show that if R is an atomic domain and a PVD, then dim(R) ≤ 1. We show that if R is a PVD and a prime ideal of R is finitely generated, then every overring of R is a PVD. We give a characterization of an atomic PVD in terms of the concept of half-factorial domain.     

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.     

The Auslander-type condition of triangular matrix rings

Let R be a left and right Noetherian ring and n,k be any non-negative integers.R is said to satisfy the Auslander-type condition G n (k) if the right flat dimension of the (i + 1)-th term in a minimal injective resolution of RR is at most i + k for any 0 i n 1.In this paper,we prove that R is Gn (k) if and only if so is a lower triangular matrix ring of any degree t over R.