Ideals of Classical Linear Lie Superalgebra P(n) over Commutative Rings

Let R be a commutative ring with identity 1. The relations between the ideals of Lie superalgebra P（n） and the ideals of R are discussed by studying the basis, center and order ideal of P（n）. All ideals of P（n） are proved to be minimal and standard.

The K_2-groups over Finaite Commutative Rings

The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.

An involutive Jordan automorphism of triangular matrix algebra over commutative rings

Let T n+1(R)be upper matrix algebra of order n+1over a 2-torsion free commutative ring Rwith identity. In this paper, we find an automorphism, which is fixed by all orthogonal idempotents and is not an R-algebraautomorphism, of T n+1(R). Furthermore we prove that this automorphism is an involutive Jordan automorphism of T n+1(R).

Ideal-based Zero-divisor Graphs of Non-commutative Rings

这份报纸介绍非可交换的戒指的一张理想男孩的零除数的图，表示我(R) 。我(R) 是一张指导的图。性质和图的可能的结构被学习。

A COMMUTATIVITY CONDITION FOR S-UNITAL RINGS

Let R be an s-unital ring, and we prove a commutativity theorem of R satisfying the following conditions: (l ) For each x, y∈ R,there exist bounded positive integers k =k (x,y), s=s (x,y), t =t (x,y)(where, at least one of k, s, t is not equal to 1) such

On Commuting Graph of Group Ring Z_nS_3

The commuting graph of an arbitrary ring R, denoted by Г（R）, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity and the diameter of Г（ZnS3）. We show that Г（ZnS3） is connected if and only if n is not a prime number. If Г（ZnS3） is connected then diam（Г（ZnS3）） = 3, while ifГ（ZnS3） is disconnected then every connected component of Г（ZnS3） must be a complete graph with same size, and we completely determine the vertice set of every connected component.

Artin环局部化结构与一类商环的等价刻画

设R是Artin交换环,m_(i)(1≤i≤k)是R的k个极大理想.通过Artin环的Jacobson根幂零和中国剩余定理得到R到⊕R/m_(i)^(l)的同态是一种同构关系.进一步把Artin环写成有限个局部环Re_(i)的直和,发现这些直和加项与商环R/m_(i)^(l)是同构关系,并且与局部环R_(m_(i))的也是同构关系.

A Generalization of Torsion Graph for Modules

Let R be a commutative ring with identity and M an R-module. In this paper, we relate a graph to M, say Γ(M), provided tsshat when M=R, Γ(M)is exactly the classic zero-divisor graph.

拟环（Nearring）的一个交换性条件

把郭华光在文献［１］中关于结合环的交换性定理推广到分配生成左拟环上。

关于近环(Near-ring)是可换环的几个定理

本文主要讨论了有单位元或无零因子的分配生成近环，以及满足无零因子。

Computation of the Orders of Classical Groups Over Finite Commutative Rings

In the present note, we compute the orders of several classes of classical groups over finite commutative rings. Simultaneously, using the order of GL_n, we obtain some Anzahl theorems of vector space over local rings. For any finite commutative ring R (with identity 1), R can be written as a direct product of a finite number of local rings R_i, i.e. , where R_i is a local ring, and the classical group G(R) can be written as . So to determine the cardinality |G(R)|, we must determine |G(R_i)| the cardinality of classical group G over a local ring.

On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.

Commutative Rings Whose Zero-divisor Graph Is a Proper Refinement of a Star Graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgraph of G induced on the vertex set V（G） ／ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G=Г（R）, which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G＊ has at least two connected components. We prove that the diameter of the induced graph G＊ is two if Z（R）2 ≠{0}, Z（R）3 = {0} and Gc is connected. We determine the structure of R which has two distinct nonadjacent vertices a, fl C Z（R）＊／{c} such that the ideal [N（a）N（β）]{0} is generated by only one element of Z（R）＊／{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kn with some end vertices adjacent to a single vertex of Kn.

Overgroups of classical groups over commutative rings in linear group

For a commutative ring with identity, we obtain a complete description of all overgroups of unitary groups U2nR (n ≥ 5), which include symplectic, ordinary orthogonal and standard unitary groups, in linear group GL2nR.

Minimal Non—Commutative n—Insertive Rings

Let n be an integer with |n| > 1. If p is the smallest prime factor of |n|, we prove that a minimal non-commutative n-insertive ring contains n^4 elements and these rings have five (2p+4) isomorphic classes for p = 2 (p p 2).

R_aR_b Transformation of Compound Finite Automata over Commutative Rings

Some results on RaRb transformation of compound finite automata over finite field are generalized to the case of commutative rings. Properties of RaRb transformation are discussed and applied to the inversion problem for compound finite automata.

THE STRUCTURE OF SYMPLECTIC GROUPS OVER ARBITRARY COMMUTATIVE RINGS

The structure of symplectic groups over fields has been determined completely. Several authors discussed symplectic groups over commutative rings, but they made strong restrictions on the underlying rings. In this letter we start from elementary symplectic matrices and,

THE STRUCTURE OF ORTHOGONAL GROUPS OVER ARBITRARY COMMUTATIVE RINGS

Let R be an arbitrary commutative ring, and n an integer≥3. It is proved for any ideal J of R thatEO2n（R, J）=[EO2n（R）, EO2n（J）]=[EO2n（R）, EO2n（R, J）]=[EO2n（R）, O2n（R,J）]=[O2n（R）, EO2n（R,J）].In particular, EO2n（R, J） is a normal subgroupof O2n（R）. Furthermore, the problem of normal subgroups of O2n（R） has an affirmative solution if and only if aR∩ Ann（2）=α2 Ann（2） for each a in R. In particular, if 2 is not a zero divisor in R, then the problem of normal subgroups of O2n（R） has an affirmative

Derivations of Certain Linear Lie Algebras over Commutative Rings

Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this paper, we first determine the intermediate algebras between h and b, then for such an intermediate algebra, we give an explicit description on its derivations, provided that R is a commutative ring with identity and 2 is invertible in R.

The structure of quadratic groups over commutative rings

Let(R,Λ)be a commutative form ring.We obtain a complete description of all subgroups of quadratic groups Q2n(R,Λ)which are normalized by elementary subgroups by form ideals,rather than usual ideals.For the weak sandwich theorem,we give a direct proof without using localization method.