Rings with Maximal Finite Subrings

In[1 ] and[2 ] ,itwas proved thata ring with a finite maximal subring mustbe finite.Special cases of this result had been shown for commutative ring(see[3 ] ) and for PI-rings (see [4 ] ) . Lanski[5] extends the result to the ring with a finite maximal invariantsubring with respect to certain automorphism.In this paper,we shall consider ringswith maximal finite subrings. Such a ring,of course,is not necessarily finite. Becauseevery finite ring has maximal finite subrings,we shall be interest…     

矩阵环的有限乘法子群

将Minkowski关于有限整数矩阵群的著名结果推广到一般的环上,主要结果是证明了:对任意环R,如果R的加法群为有限生成的自由Abel群,则R的所有乘法可逆元构成的群U(R)中的有限子群精确到同构只有有限多个.     

Generalized Iterated Maximal Essential Extensions of Rings

A necessary and sufficient condition for the existence of GIME(A), the generalized iterated maximal essential extension of an associative ring A, is obtained provided that A is commutative. Some properties of GIME(A) are described, in particular, when A is semiprime or commutative. An example of a prime ring A having no GIME(A) is constructed.2000 Mathematics Subject Classification: 16S70, 13G05, 13E05, 13B02, 13A15     

一类有限环

本文指出了文 [5 ]的一些错误 ,给出了文 [5 ]定理 1的一个反例 .并给出了一类交换环的等价条件 .     

关于有向有限环的某些结果

一个环R称为有向有限的,如果对于x,y∈R,xy=1蕴涵着yx=1.本文我们首先建立有向有限环的某些新的刻画,然后考察了它们的某些性质.     

Indices of Maximal Subgroups of Finite Soluble Groups

We look at the structure of a soluble group G depending on the value of a function m(G)= max m _p G), where m _p(G)=max{log_p|G:M| | M< G, |G:M|=p ^a}, p (G). Theorem 1 states that for a soluble group G, (1) r(G/ (G))= m(G); (2) d(G/ (G)) 1+ (m(G)) 3+m(G); (3) l _p(G) 1+t, where 2^t-1<m _p(G) 2^t. Here, (G) is the Frattini subgroup of G, and r(G), d(G), and l _p(G) are, respectively, the principal rank, the derived length, and the p-length of G. The maximum of derived lengths of completely reducible soluble subgroups of a general linear group GL(n,F) of degree n, where F is a field, is denoted by (n). The function m(G) allows us to establish the existence of a new class of conjugate subgroups in soluble groups. Namely, Theorem 2 maintains that for any natural k, every soluble group G contains a subgroup K possessing the following properties: (1) m(K); k; (2) if T and H are subgroups of G such that K T <_max <_max H G then |H:T|=p ^t for some prime p and for t>k. Moreover, every two subgroups of G enjoying (1) and (2) are mutually conjugate.     

Generalized PP and Zip Subrings of Matrix Rings

Let R be an abelian ring. We consider a special subring An, relative to α2,…, αn∈ REnd（R）, of the matrix ring Mn（R） over a ring R. It is shown that the ring An is a generalized right PP-ring （right zip ring） if and only if the ring R is a generalized right PP-ring （right zip ring）. Our results yield more examples of generalized right PP-rings and right ziu rings.     

Finite Groups with 30 Elements of Maximal Order

It is an interesting topic to determine the structure of a finite group which has a given number of elements of maximal order. In this article, the author classified finite groups with 30 elements of maximal order. This work is supported by NSFC (No. 10571128), NSF of CHONGQING (CSTC: 2005BB8096).     

Rings whose nilpotents form a multiplicative set

Nilpotents in Armendariz and abelian π-regular rings are multiplicatively closed. However, this fact need not hold in many kinds of rings. This article concerns a class of rings whose nilpotents are closed under multiplication. Rings contained in this class are said to be nilpotent-closed, and the structure of nilpotent-closed rings is investigated in relation with various situations which happen ordinarily in the study of noncommutative ring theory. In the procedure, various sorts of rings are investigated, so that they are nilpotent-closed. We also study familiar conditions under which nilpotents in nilpotent-closed rings form a subring.     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

有限交换群中的极大迷向子模

设有限群G作用在有限交换群E上,并且E带有一个G-不变的乘法辛型.如果U和V为E的两个极大迷向的G-子模,证明了相伴的两个辛G-模U⊥/U和V⊥/V同构.     

有限结合环类中的特殊根

设Ω是有限结合环类中全部弱单环组成的环类，Ω１∪Ω２＝Ω，Ω１∩Ω２＝Φ，在有限结合环类中，我们证明了ＬΩ１＝ＵΩ２可以成立，并给出等式成立的充要条件．使用这个结论，我们可以证明，在有限结合环类中，超幂零根是特殊根．     

Flat Modules over Group Rings of Finite Groups

Let k be a commutative ring of coefficients and G be a finite group. Does there exist a flat k G-module which is projective as a k-module but not as a k G-module? We relate this question to the question of existence of a k-module which is flat and periodic but not projective. For either question to have a positive answer, it is at least necessary to have |k| ≥ ?_ω. There can be no such example if k is Noetherian of finite Krull dimension, or if k is perfect.     

On F-Abnormal Maximal Subgroups of Finite Groups

For any saturated formation F of finite groups containing all supersolvable groups, the groups in F are characterized by F-abnormal maximal subgroups.     

Finite Rank Operators in Closed Maximal Triangular Algebras

Ｆｉｎｉｔｅｒａｎｋｏｐｅｒａｔｏｒｓａｎｄｒａｎｋｏｎｅｏｐｅｒａｔｏｒｓａｒｅｉｍｐｏｒｔａｎｔｔｏｔｈｅｔｈｅｏｒｙｏｆｎｅｓｔａｌｇｅｂｒａｓ .Ｉｎａｎｅｓｔａｌｇｅｂｒａ ,ｅａｃｈｆｉｎｉｔｅｒａｎｋｏｐｅｒａｔｏｒｃａｎｂｅｗｒｉｔｔｅｎａｓａｆｉｎｉｔｅｓｕｍｏｆｒａｎｋｏｎｅｏｐｅｒａｔｏｒｓｗｈｉｃｈｂｅｌｏｎｇｔｏｉｔｓｅｌｆ (Ｔｈｉｓｒｅｓｕｌｔｉｓｉｎ [1 ],ｂｕｔｂｅｌｏｎｇｓｔｏＪ .Ｒ .Ｒｉｎｇｒｏｓｅ) .Ｉｎａｎａｔｕｒａｌｗａｙ ,ｗｅｍａｙａｓｋ :　Ｑｕｅｓｔｉｏｎ…     

Maximal submodules and locally perfect rings

Rings over which every nonzero right module has a maximal submodule are calledright Bass rings. For a ringA module-finite over its centerC, the equivalence of the following conditions is proved:

Covering dimension and finite spaces

Finite topological spaces, that is spaces with a finite number of points, have a wide range of applications in many areas such as computer graphics and image analysis. In this paper we study the covering dimension of a finite topological space. In particular, we give an algorithm for computing the covering dimension of a finite topological space using matrix algebra.     

Rings with divisibility on chains of ideals

We present a generalization of the ascending and descending chain condition on one-sided ideals by means of divisibility on chains. We say that a ring R satisfies ACC_d on right ideals if in every ascending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is a left multiple of the following one; that is, each right ideal in the chain, except for a finite number, is divisible by the following one. We study these rings and prove some results about them. Dually, we say that a ring R satisfies DCC_d on right ideals if in every descending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is divisible by the previous one. We study these conditions on rings, in general and in special cases.     

非幂零极大子群共轭类类数给定的有限群

证明了非幂零极大子群共轭类类数等于2的有限群必可解,并给出了非幂零极大子群同阶类类数等于2的非可解群的等价刻画.     

Quasi-Baer Rings with Essential Prime Radicals

A ring R is called “quasi-Baer” if the right annihilator of every right ideal is generated, as a right ideal, by an idempotent. It can be seen that a quasi-Baer ring cannot be a right essential extension of a nilpotent right ideal. Birkenmeier asked: Does there exist a quasi-Baer ring which is a right essential extension of its prime radical? We answer this question in the affirmative. Moreover, we provide an example of a quasi-Baer ring in which the right essentiality of the prime radical does not imply the left essentiality of the prime radical.