ON ORDER RINGS OF SEMI-PRIMARY RINGS

The main results of this paper are stated as follows.Let R be an orderring in thesemi-primary ring Q.Suppose that R satisfies the maximal condition for nil right ideals ofR,Then we have(i)if an ideal I of R has a finite length as right R-module,then I alsohas a finite length as left R-module;(ii)denote by A(R)the Artinian radical of R,andN the nil radical of R,then A(R)+N/N=A(R/N),if R satisfies the commutative condi-tion on the zero product of prime ideals of B.     

ISOMORPHISMS OF CHARACTER RINGS

Let G and H be finite groups. It is proves that any ring isomorphism betweencharacter rings of G and H implies a perfect isometry between them, which is also a ringisomorphism.     

GRADED MODULES OF GRADED LIE ALGEBRAS OF CARTAN TYPE (Ⅲ)——IRREDUCIBLE MODULES

Over a field of characteristic≠2, 3, all irreducible positive and negative graded modulesof simple Lie algebras L(n) and L(n, m) of Cartan type W, S and H are determined.Further, all irreducible positive and negative filtered modules of L(n, m) are determined.For L,(n), every irreducible negative filtered module is a negative graded module, but thereexist irreducible positive filtered modules which are not graded.     

ON THE STRUCTURE OF PRIMITIVE RINGS

In this paper the author introduces two concepts, i. e. the concept of so-colled v-socles of primitive rings and the concept of a pair of dual modules. Then the author establishes a general structure theorem for primitive rings with v-socles, which implies the well-known structure theorem for primitive rings with usual non-zero socles.     

GRADED MODULES OF GRADED LIE ALGEBRAS OF CART AN TYPE (III) —IRREDUCIBLE MODULES

Over a field of characteristic$\[ \ne 2\]$, 3, all irreducible positive and negative graded modules of simple Lie algebras $\[L(n)\]$ and $\[L(n,m)\]$ of Cartan typs $\[W,S\]$, and H are determined. Further, all irreducible positive and negative filtered modules of $\[L(n,m)\]$ are determined.For $\[L(n)\]$, every irreducible negative filtered module is a negative graded module, but there exist irreducible positive filtered modules which are not graded.     

STABILIZATION OF UNITARY GROUPSOVER POLYNOMIAL RINGS

ＳＴＡＢＩＬＩＺＡＴＩＯＮＯＦＵＮＩＴＡＲＹＧＲＯＵＰＳＯＶＥＲＰＯＬＹＮＯＭＩＡＬＲＩＮＧＳ￥ＹＯＵＨＯＮＧＡｂｓｔｒａｃｔ：Ｔｈｅａｕｔｈｏｒｓｔｕｄｉｅｓｔｈｅｓｔａｂｉｌｉｚａｔｉｏｎｆｏｒｔｈｅｕｎｉｔａｒｙｇｒｏｕｐｓｏｖｅｒｐｏｌｙｎｏ...     

HOMOLOGICAL PROPERTIES OF TORSION CLASSES UNDER CHANGE OF RINGS

ＨＯＭＯＬＯＧＩＣＡＬ ＰＲＯＰＥＲＴＩＥＳ ＯＦ ＴＯＲＳＩＯＮ ＣＬＡＳＳＥＳ ＵＮＤＥＲ ＣＨＡＮＧＥ ＯＦ ＲＩＮＧＳ ￥ＹＡＯＭＵＳＨＥＮＧＡｂｓｔｒａｃｔ：ＬｅｔＲｂｅａｒｉｎｇｗｉｔｈｉｄｅｎｔｉｔｙ，ｘｂｅａｃｅｎｔｒａｌｅｌｅｍｅｎ...     

THE INVARIANT FORMS ON THE GRADED MODULES OF THE GRADED CARTAN TYPE LIE ALGEBRAS

This paper determines the graded modules of graded Cartan type Lie aigebras whichpossess nondegenerate invariant form.     

SPECTRAL PROPERTIES OF POPULATION OPERATOR AND ASYMTOTIC BEHAVIOUR OF POPULATION SEMIGROUP

In this paper we discuss the spectral properties of the Population operator, prove that the population operator has only one real eigenvalue γ_0, which is greater that real parts of other eigenvalues,and find the corresponding relation between γ_0 and the critical fertility rate β_(cr). We also study the existence and asymptotic behaviour of the semigroup for the population system, then come to the conclusion about the stability of the population system,including the existene of the steaby population state in the critical case of the fertility rate.These are all the new results of the quantitative population theory.     

A NOTE ON QUASI-P RADICAL OF ASSOCIATIVE RINGS

Let P be an arbitrary property of rings and homomorphlcally closed. For an arbitraryring A we construct quasi-P radical O.P(A) with transfinite induction, and give another character-ization of quasi-P radical;     

ON THE DIFFERENTIABILITY OF THE PARITY PROGRESSIVE POPULATION SEMIGROUP

ＯＮＴＨＥＤＩＦＦＥＲＥＮＴＩＡＢＩＬＩＴＹＯＦＴＨＥＰＡＲＩＴＹＰＲＯＧＲＥＳＳＩＶＥＰＯＰＵＬＡＴＩＯＮＳＥＭＩＧＲＯＵＰ￥ＳＨＩＤＥＭＩＮＧＡＮＤＹＡＮＧＬＵＳＨＡＮ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＺｈｅｎｇｚｈｏｕＵｎｉｖｅ...     

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 thatEO_(2n)(R, J)=[EO_(2n)(R), EO_(2n)(J)]=[EO_(2n)(R), EO_(2n)(R, J)]=[EO_(2n)(R), O_(2n)(R,J)]=[O_(2n)(R), EO_(2n)(R,J)].In particular, EO_(2n)(R, J) is a normal subgroupof O_(2n)(R). Furthermore, the problem of normal subgroups of O_(2n)(R) has an affirmative solution if and only if aR∩ Ann(2)=α~2 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 O_(2n)(R) has an affirmative solution     

ALGEBRAIC GROUPS ASSOCIATED WITH GRADED LIE ALGEBRAS OF CARTAN TYPE

§０．ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｋｎｏｗｎｔｈａｔｏｖｅｒａｌｇｅｂｒａｉｃａｌｙｃｌｏｓｅｄｆｉｅｌｄｓｏｆｃｈａｒａｃｔｅｒｉｓｔｉｃ０，ｔｈｅｒｅｅｘｉｓｔｓａｇｏｏｄｃｏｒｅｓｐｏｎｄｅｎｃｅｂｅｔｗｅｅｎｔｈｅｃｏｎｎｅｃｔｅｄ...     

COHOMOLOGY OF GRADED LIE ALGEBRAS OF CARTAN TYPE S (n, m)

Let F be an algebraically closed field of characteristic p>3. This paper studies the cohomology of graded Lie algebras S(n,m) of Cartan type over F. The author determines the structures of H~1(S(3,m), _0), where _0 is the graded S(3,m)-module which is induced by an irreducible sl (3)-module V_0, the structures of H~1(S(3,m), V), where V is an irreducible S(3,m)-module, and the structures of the restricted cohomology H_*~1(S(3,(1, 1, 1)), V), where V is an irreducible restricted S(3, (1, 1, 1))-module.     

ON THE REACHABLE SEMIGROUP OF BILINEAR CONTROL SYSTEMS ON LIE GROUP

This paper studies the reachability and the structure of reachable semigroup of bilinear control systems on Lie group. In the second section some equivalency lemmas are given, which not only simplify the proofs of the main results, but discover some properties of systems also. In the third section some conditions are advanced that the reachable semigroup of system is weakly symmetric by means of the study of one prameter subgroups. This study is discussed by manifold theory and matrix theory, respectively. In the last seetion, some topological properties of the reachable semigroup are advanced.     

*—BI—IDEALS IN RADICAL THEORY OF INVOLUTION RINGS

Abstract

If a radical class is closed under involution in every ring with involution then the radical theoretic conditions involving *-bi-ideals are equivalent to the corresponding conditions concerning bi-ideals without involution.     

A CHARACTERIZATION OF Mπ-GROUPS

Let π be a set of primes. Isaacs established the π-theory of characters, which generalizes the theory of Brauer modular characters. Motivated by Isaacs＇s work, we introduce the definition of Mπ-groups and provide a characterization of Mπ-groups.     

ON THE QUOTIENT RING OF COMMUTATIVE RINGS WITH ACC⊥ON ANNIHILATOR IDEALS

The author concludes that every commutative ring with ascending chain conditionan annihilator ideals has a Kasch quotient ring,which generalizes the Theorem thatevery commutative noetherian ring has a Kasch quosient ring.If follows that if R is acommutative ring with acc~⊥,then that Q(R)is semiprimary is equivalent to that it isperfect,or to that R satisfies regular condition.Besides,that Q(R)is quasi-frobeniusequals that Q(R)is FPF or PF,and that Q(R)is artinian equals that B/N,are of finitedimension,i=1,2,…,n.N_i,=J~i∩R.