半群根的根补及其若干性质

给出理想遗传根的根补根半群被刻划，并证明了半群根的根补的若干性质。     

Supplementing Radicals and Decompositions of Near-Rings

If is a radical of near-rings and is its supplementing radical, then (N)(N) N. We address the issue when (N) (N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N N_c is a hereditary Kurosh–Amitsur radical, _c is characterized in terms of distributors and criteria are given for the decomposition N = c(N) c(N). In the subvariety A of all abstract affine near-rings, assigning the maximal torsion ideal (N) is a hereditary Kurosh–Amitsur radical. If such near-rings N A satisfy dcc on principal right ideals, then N splits into a direct sum N = (N) (N) where the additive group of (N) is torsionfree and divisible. Dropping dcc on principal right ideals, an ``essential" decomposition result is proved.     

群的遗传根性和强半单根性

本文利用群的根性的性质,解决了Szasz在环的根性理论中提出的公开问题在群论中的对应问题.同时我们研究了群的遗传根性和强半单根性的一些性质,并介绍了群的根类的交运算和并运算,由此得到了一些很好的结果.     

Lattices of Radicals of Involution Rings

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We show that certain sublattices are isomorphic to sublattices of the lattice of radicals of rings. We characterize the atoms of certain lattices of radicals of involution rings.1991 Mathematics Subject Classification: 16W10, 16N99, 06B99     

The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical

Let G be a finite group, and let p ₁,…, p _m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P ₁,…, P _m , where P _i is a Sylow p _i -subgroup of G, and g ∈ G, denote by m _𝒫(g) the number of factorizations g = g ₁…g _m such that g _i  ∈ P _i . Previously, it was shown that the properly normalized average of m _𝒫 over all 𝒫 is a complex character over G termed the Average Sylow Multiplicity Character. The present article identifies the kernel of this character as the subgroup of G consisting of all g ∈ G such that m _𝒫(gx) = m _𝒫(x) for all 𝒫 and all x ∈ G. This result implies a close connection between the kernel and the solvable radical of G.     

代数正规类中的遗传根与强半单根

Puczylowski建立了一般代数对象类的根理论.本文在代数正规类中,用格论方法刻划一般遗传根和强半单根类,探究它们的一些性质,推广了已知各类代数系统的某些根论研究.     

On a Hereditary Radical Property Relating to the Reducedness

In this note, a hereditary radical property, called homomorphically reduced rings, is introduced, observed, and applied. The dual concept of this property is also studied with the help of Courter, proving that any ring R (possibly without identity) has an ideal S such that S/K is not homomorphically reduced for each proper ideal K of S; and if L is an ideal of R with L ? S, then L/H is homomorphically reduced for some ideal H of R with H ? L. The concept of the homomorphical reducedness is shown to be equivalent to the left (right) weak regularity and the (strong) regularity for one-sided duo rings. It is proved that homomorphically reduced rings have several useful properties similar to those of (weakly) regular rings. It is proved that the homomorphical reducedness can go up to classical quotient rings. It is shown that if R is a reduced right Ore ring with the ascending chain condition (ACC) for annihilator ideals, then the maximal right quotient ring of R is strongly regular (hence homomorphically reduced).     

Note on Direct Products of Certain Classes of Finite Groups

Direct products of solvable groups in which Sylow-permutability is a transitive relation are analyzed. Such groups are called 𝒫𝒮𝒯 groups, and it is well known that if the orders of two 𝒫𝒮𝒯 groups are relatively prime, then their direct product is again a 𝒫𝒮𝒯 group. Examples suggest that for solvable groups it is not necessary to have relatively prime orders to stay in the class. Some characterizations are provided, and also direct products of some other classes are discussed.     

分次反单本原根

本文给出分次情形的反单本原根SJGG,当群G满足|G|<∞,证明了SJG=SJref=SJG,其中SJref和SJG分别表示反射反单本原根和限制反单本原根.而且讨论了SJG的分次补根.     

On Radicals with Amitsur Property

Abstract

A radical γ has the Amitsur property, if γ(A[x]) = (γ(A[x]) ∩ A)[x] for every ring A. To any radical γ with Amitsur property we construct the smallest radical γ ^x which coincides with γ on polynomial rings. Distinct special radicals with Amitsur property are given which coincide on simple rings and on polynomial rings, answering thus a stronger version of M. Ferrero's problem. Radicals γ with Amitsur property are characterized which satisfy A[x, y] ∈ γ whenever A[x] ∈ γ.     

On the composition subalgebras of Bridgeland’s Hall algebras

Let A be a finite dimensional hereditary algebra over a finite field k and 𝒫 the category consisting of finite dimensional projective (left) A-modules. In this paper, we consider the composition subalgebra of Bridgeland’s Hall algebra of the category 𝒞_m(𝒫) of m-cyclic complexes for any positive integer m≥2 and determine its generating relations.     

Some new dimensions of modules and rings

We say that a class 𝒫 of right modules over a fixed ring R is an epic class if it is closed under homomorphic images. For an arbitrary epic class 𝒫, we define a 𝒫-dimension of modules that measures how far modules are from the modules in the class 𝒫. For an epic class 𝒫 consisting of indecomposable modules, first we characterize rings whose modules have 𝒫-dimension. In fact, we show that every right R-module has 𝒫-dimension if and only if R is a semisimple Artinan ring. Then we study fully Hopfian modules with 𝒫-dimension. In particular, we show that a commutative ring R with 𝒫-dimension (resp. finite 𝒫-dimension) is either local or Noetherian (resp. Artinian). Finally, we show that Mat_m(R) is a right Köthe ring for some m if and only if every (left) right module is a direct sum of modules of 𝒫-dimension at most n for some n, if and only if R is a pure semisimple ring.     

Graded Antisimple Primitive Radical

We introduce the graded version of the antisimple primitive radical $ {\user1{\mathcal{S}\mathcal{J}}} $ , the graded antisimple primitive radical $ {\user1{\mathcal{S}\mathcal{J}}}_{G} $ . We show that $ {\user1{\mathcal{S}\mathcal{J}}}_{G} = {\user1{\mathcal{S}\mathcal{J}}}_{{{\text{ref}}}} = {\user1{\mathcal{S}\mathcal{J}}}^{G} $ when |G| < ∞, where $ {\user1{\mathcal{S}\mathcal{J}}}_{{{\text{ref}}}} $ denotes the reflected antisimple primitive radical and $ {\user1{\mathcal{S}\mathcal{J}}}^{G} $ denotes the restricted antisimple primitive radical. Furthermore, we discuss the graded supplementing radical of $ {\user1{\mathcal{S}\mathcal{J}}}^{G} $ .     

Algebraic quantum kk-theory

Let 𝒢 be an algebraic quantum group. We introduce an equivariant algebraic kk-theory for 𝒢-module algebras. We study an adjointness theorem related with smash product and trivial action. We also discuss a duality property.     

分次根与自反根

Ｍ．Ｂｅａｔｔｉｅ和Ｐ．Ｓｔｅｗａｒｔ在文（１）中讨论了六种分次根与相应自反根的关系。作为该文的继续，本文首先定义了Ｋｏｅｔｈｅ根并讨论它与Ｋｏｅｔｈｅ自反根的关系。其次，进一步给出Ｌｅｖｉｔｚｋｉ自反根的刻画。最后，我们指出文（１）的两个错误结论，证明分次ｖｏｎ Ｎｅｕｍａｎｎ根与ｖｏｎ Ｎｅｕｍａｎ自反根恰好相等。     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

Exceptional Pairs in Hereditary Categories

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ?. If r = dim _k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H _r ) over the r-Kronecker algebra H _r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ? is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ?¹ or to mod(H _r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.     

On sums and products of primitive elements

Let R be a (commutative integral) domain, with K its quotient field and R^′ its integral closure (in K). Let 𝒫 be the set of elements u∈K such that u is primitive over R; i.e., such that u is the root of a polynomial over R having a unit coe?cient. Then, 𝒫 is a ring (necessarily K) ? 𝒫 is closed under products ? R^′ is a Prüfer domain. In general, 𝒫 is closed under powers. For u,v∈𝒫, necessary and su?cient conditions are given for u+v (resp., uv) to belong to 𝒫. Also, 𝒫 is used to characterize when R is a quasi-local integrally closed domain and when R is a pseudo-valuation domain. If R is quasi-local, each element of K is expressible as the sum of two (possibly equal) elements of 𝒫. The set of primitive elements is determined for lying-over pairs and for extensions of domains with the same sets of prime ideals. In this study of the 𝒫 construction, R and K are replaced, whenever possible, by an arbitrary commutative ring and its total quotient ring or, more generally, by any inclusion of commutative rings.