质环的求导和交换性

In this paper, we generalize some corresponding results of [1-4]. We obtain the main results as the following:Theorem 1 Let R be a prime ring of characteristic not 2 with nontrivial derivations d₁,d₂ and let U be a nonzero ideal of R . If C is the center of R, then the following conditions are equivalent : (i)d₁,d₂(x)∈C for all x∈U; (ii) [d₁(x),d₂(y)]∈C for all x,y∈U; (iii) d₁(x)d₂(y)+d₂(x)d₁(y)∈C for all x,y∈U; (iv) R is commutative.Theorem 2 Let R be a prime ring with nontrivial derivations d₁,d₂,…, d_n and U be a nonzero ideal of R. Let C be the center of R. If d₁(x₁)d₂(x₂)…d_n(x_n)∈C for all x₁, x₂…x_n)∈U, then R is commutative.     

反向极限与模的FP-内射包

Let R be a ring with unit element, and A a left R-module. A is called FP-injective if Ext_R′(P, A)=0 for every finitely presented R-module P. Let E_R(A) denote the injective hull of A. In the paper we prove by means of inverse limit functor that any module A Over coherent ring R has the minimum FP-injective submodule e_R(A) in E_R(A) which contains A and can be defined as the FP-injective hull of A.     

PF环与群环的Grothendieck群

Let R be a commutative ring with 1, G an Abelian group, RG the group ring on R and G. In this paper we gave some properties of PF- rings in which f. g. projective modules are free. The Grothendieck groups K₀(RG) for some cases are given. In addition, for the ring R with the unimodular column property, we proved the following result: K₀(RG) ≈K₀(R), hence if R ∈PF, then K₀(RG)≈Z .     

Cohn环与Grothendieck群

We introduce three classes of cohn rings C₁,C₂ and C₃). Let R and S are rings,φ : R→S is a ring homomorphism. We prove that R∈C_i if and only if R/J(R) ∈ C_i. In We also discuss the relation between the     

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

In this paper it is proved that local fundamental solution exists in some space W^m(H_n) (m∈Z), if the left invariant differential operator on the Heisenberg group H_n satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on H_n. If there exist R≥0, r,s∈R and operators {B_λ|λ∈Γ_R} ∈V^s(Γ_R, M^r) such that, for almost all λ∈Γ_R, B_λ is the right inverse of Ⅱ_λ(L), then there exists E∈W^m(H_n) (when m≥0 or m even) or E∈W^m-1(H_n) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and C_λ≥ C|λ|^x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2).     

范畴RMnl上的一个函子

Let _RM_n^l′ be a category which is equivalent to the category of left R-modules.In this paper,we define afunction F:_RM_n^l→_RM_n^l′ and prove that the functor Fpreserves products,direct limits,injections,surjectios and total esactness.Finally,we show that the functor F is a left-adjoint of the inclusion functor I:_RM_n^l′→_RM_n^l. Hence I:_RM_n^l′ is a renective subcategory of _RM_n^l.     

正则环的内射理想

设R为正则环, A∈ M_n(R). 如果M_n(R)AM_n(R)为M_n(R) 的内射理想,该文证明了有U, V∈ GL_n(R)使得UAV为对角矩阵.     

关于Johns的经验Bayes估计的几点注记

Some problems related to the empirical Bayers estimate proposed by Johns[1] (δ_n), and its modification (δ_n′) by Maritz[3], are considered in this artical. The main result concerns the comparison of the remainders R(G,δ_n) - R(G) and R(G,δ_n′)-R (G), where R(G,δ_n) and R(G,δ_n′) denote the "over-all" Bayes risk(under quadratic loss) of δ_n and δ_n′ respectively, and R(G) denotes the Bayes risk of Bayes estimate, and G is the prior distribution.     

正则环和GP-内射环

A ring R is called left Gp-injective if for any a∈R, there exists a positive integer n such that any left R-homomorphism of Raⁿ into R extends to one of R into R. In this paper, we prove that the centre of semiprime (left nonsingular) left GP- injective ring is regular ring, and improve some propositions in [3].     

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × R*

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mⁿ × R, where Mⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.