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.

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.

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.

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.

ISOMORPHISMS OF GL_3 OVER COMMUTATIVE RINGS

In this paper, we discuss the necessary and sufficient conditions in which there exist exceptional isomorphisms between E3（A） and E3（R） and between GL3（A） and GL3（R） for commutative rings A and R. A counterexample shows that two rings that are not isomorphic may have isomorphic tinear groups. This partly solves an open problem posed at the conference on abstract homomorphisms of algebraic groupsrs.

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.

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).

One Kind of Complete Lie Algebra over a Commutative Ring

In this paper, we obtain a new kind of complete Lie algebra over a commutative ring, which is the Lie algebra consisting of all n × n anti-symmetric matrices over a 2-torsionfree commutative ring with identity.

MATRIX PRODUCT CODES WITH ROSENBLOOM-TSFASMAN METRIC

In this article, the Rosenbloom-Tsfasman metric of matrix product codes over finite commutative rings is studied and the lower bounds for the minimal Rosenbloom- Tsfasman distances of the matrix product codes axe obtained. The lower bounds of the dual codes of matrix product codes over finite commutative Frobenius rings are also given.

Higher Level Orderings on Modules

The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of higher level orderings are generalized to thecategory of modules over commutative rings. Moreover, a strict intersection theorem for higherlevel orderings on modules is established.

Finite Rings Whose Graphs Have Clique Number Less than Five

Let R be a commutative ring and Γ(R)be its zero-divisor graph.We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one,two,or three.Furthermore,if R■R1×R2×…×Rn(each Ri is local for i=1,2,3,...,n),we also give algebraic characterizations of the ring R when the clique number of Γ(R)is four.

On Finite Local Rings with Clique Number Four

We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R)is a commutative regular ring with J(R)nil,where J(R)is the Jacobson radical of R.

The Classification of Torsion-free TI-Groups

An abelian group A is called a TI-group if every associative ring with the additive group A is filial.The filiality of a ring R means that the ring R is associative and all ideals of any ideal of R are ideals in R.In this paper,torsion-free TI-groups are described up to the structure of associative nil groups.It is also proved that,for torsion-free abelian groups that are not associative nil,the condition TI implies the indecomposability and homogeneity.The paper contains constructions of 2^(■o)such groups of any rank from 2to 2^(■o)which are pairwise non-isomorphic.

On a New Extension of the Zero-Divisor Graph

In this paper,we introduce a new graph whose vertices are the non-zero zerodivisors of a commutativc ring R、and for distincts elements x and y in the set Z(R)^(*)of the non-zero zero-divisors of R,x and y are adjacent if and only if xy=0 or x+y∈Z(R).We present some properties and examples of this graph,and we study its relationship with the zero-divisor graph and with a subgraph of the total graph of a commutative ring.

On the Number Counting of Polynomial Functions

Polynomial functions （in particular, permutation polynomials） play an important role in the design of modern cryptosystem. In this note the problem of counting the number of polynomial functions over finite commutative rings is discussed. Let A be a general finite commutative local ring. Under a certain condition, the counting formula of the number of polynomial functions over A is obtained. Before this paper, some results over special finite commutative rings were obtained by many authors.

Normalization of Indexed Differentials by Extending Grobner Basis Theory

It is a fundamental problem to determine the equivalence of indexed differential polynomials in both computer algebra and differential geometry.However,in the literature,there are no general computational theories for this problem.The main reasons are that the ideal generated by the basic syzygies cannot be finitely generated,and it involves eliminations of dummy indices and functions.This paper solves the problem by extending Grobner basis theory.The authors first present a division of the set of elementary indexed differential monomials E■ into disjoint subsets,by defining an equivalence relation on E■ based on Leibniz expansions of monomials.The equivalence relation on E■also induces a division of a Grobner basis of basic syzygies into disjoint subsets.Furthermore,the authors prove that the dummy index numbers of the sim-monomials of the elements in each equivalence class of E■ have upper bounds,and use the upper bounds to construct fundamental restricted rings.Finally,the canonical form of an indexed differential polynomial proves to be the normal form with respect to a subset of the Grobner basis in the fundamental restricted ring.

An Extension of Grobner Basis Theory to Indexed Polynomials Without Eliminations

In computer algebra,it remains to be challenging to establish general computational theories for determining the equivalence of indexed polynomials.In previous work,the author solved the equivalence determination problem for Riemann tensor polynomials by extending Grobner basis theory.This paper extends the previous work to more general indexed polynomials that involve no eliminations of indices and functions,by the method of ST-restricted rings.A decomposed form of the Grobner basis of the defining syzygy set in each ST-restricted ring is provided,and then the canonical form of an indexed polynomial proves to be the normal form with respect to the Grobner basis in the ST-fundamental restricted ring.