The connes spectrums of certain finite non-abelian groups

In this article, we investigate when every simple module has a projective (pre)envelope. It is proven that (1) every simple right R-module has a projective preenvelope if and only if the left annihilator of every maximal right ideal of R is finitely generated; (2) every simple right R-module has an epic projective envelope if and only if R is a right PS ring; (3) Every simple right R-module has a monic projective preenvelope if and only if R is a right Kasch ring and the left annihilator of every maximal right ideal of R is finitely generated.     

ON SOME PROPERTIES OF IF RINGS

Abstract

We show that left IF rings (rings such that every injective left module is flat) have certain regular-like properties. For instance, we prove that every left IF reduced ring is strongly regular. We also give characterizations of (left and right) IF rings. In particular, we show that a ring R is IF if and only if every finitely generated left (and right) ideal is the annihilator of a finite subset of R.     

广义FP—内射模、广义平坦模与某些环

左（右）R－模A称为GFP－内射模，如果ExtR(M,A)=0对任－2－表现R－模M成立；左（右）R－模称为G－平坦的，如果Tor1^R(M,A)=0(Tor1^R(AM)=0)对于任一2－表现右（左）R－模M成立；环R称左（右）R－半遗传环，如果投射左（右）R－模的有限表现子模是投射的，环R称为左（右）G－正而环，如果自由左（右）R－模的有限表现子模为其直和项，研究了GFP－内射模和G－平坦模的一些性质，给出了它们的一些等价刻划，并利用它们刻划了凝聚环，G－半遗传环和G－正则环。     

关于$(m,n)$-凝聚模与预包络

In this paper, let m, n be two fixed positive integers and M be a right R-module, we define （m, n）-M-flat modules and （m, n）-coherent modules. A right R-module F is called （m, n）-M-flat if every homomorphism from an （n, m）-presented right R-module into F factors through a module in addM. A left S-module M is called an （m, n）-coherent module if MR is finitely presented, and for any （n, m）-presented right R-module K, Hom（K, M） is a finitely generated left S-module, where S = End（MR）. We mainly characterize （m, n）-coherent modules in terms of preenvelopes （which are monomorphism or epimorphism） of modules. Some properties of （m, n）-coherent rings and coherent rings are obtained as corollaries.     

Gorenstein Projective Precovers

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.     

A generalization of right pci rings

By a well-known result of Osofsky [6, Theorem] a ring R is semisimple (i.e. R is right artinian and the Jacobson radical of R is zero) if and only if every cyclic right R-module is injective. Starting from this, a larger class of rings has been introduced and investigated, namely the class of right PCI rings. A ring R is called right PCI if every proper cyclic right R- module is injective (proper here means not being isomorphic to RR). By [l] and [Z], a right PCI ring is either semisimple or it is a right noetherian, right hereditary simple ring. The latter ring is usually called a right PCI domain. In this paper we consider the similar question in studying rings whose cyclic right modules satisfy some decomposition property. The starting point is a theorem recently proved in 13, Theorem 1.1): A ring R is right artinian if and only if every cyclic right R- module is a direct sum of an injective module and a finitely cogenerated module.     

关于(m,n)-凝聚模与预包络

In this paper,let m,n be two fixed positive integers and M be a right R-module,we define (m,n)-M-flat modules and (m,n)-coherent modules.A right R-module F is called (m,n) M-flat if every homomorphism from an (n,m)-presented right R-module into F factors through a module in addM.A left S-module M is called an (m,n)-coherent module if MR is finitely presented,and for any (n,m)-presented right R-module K,Horn(K,M) is a finitely generated left S-module,where S = End(MR).We mainly characterize (m,n)-coherent modules in terms of preenvelopes (which are monomorphism or epimorphism) of modules.Some properties of (m,n)-coherent rings and coherent rings are obtained as corollaries.     

PI-rings and semiprime rings having faithful modules with krull dimension

It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent. If, moreover, the R-module M and the left ideal_R(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings. Moreover, first, a right Artinian ring R such that rad(R)²=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension. Supported by RFFR grant No. 26-93-011-1544. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 562–572, September–October, 1997.     

On（N,U）-Coherence of Modules and Rings

Let U be a flat right R-module and N an infinite cardinal number.A left R-module M is said to be (N,U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (N,U)-finitely presented in σ[M].It is proved under some additional conditions that a left R-module M is (N,U)-coherent if and only if Л^Ni∈I U is M-flat as a right R-module if and only if the (N,U)-coherent dimension of M is equal to zero.We also give some characterizations of left (N,U)-coherent dimension of rings and show that the left N-coherent dimension of a ring R is the supremum of (N,U)-coherent dimensions of R for all flat right R-modules U.     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves     

On Covers and Envelopes in Some Functor Categories

We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every left R-module M has an Ext-phantom preenvelope f: M → N with coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by the FP-injective objects.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

Excellent Extensions of Rings

Ｅｘｃｅｌｌｅｎｔ　Ｅｘｔｅｎｓｉｏｎｓ　ｏｆ　ＲｉｎｇｓＬｉｕＺｈｏｎｇｋｕｉ（刘仲奎）ａｎｄＷａｎｇＴｉｎｇＺｈｅｎ（王廷桢）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｃｍａｔｉｃｓ，ＮｏｒｔｈｗｅｓｔＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｌａｎｚｈｏｎ，７...     

Coherent rings of finite weak global dimension

The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, Martínez). We will exploit these features of this category to study its objects. In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.

     

Self-Orthogonal Modules of Finite Projective Dimension

Let R be a ring and _R ω a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to _R ω) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module _R ω, we show that the projective dimension of _R ω and the right orthogonal dimension (relative to _R ω) of R/J are identical, where J is the Jacobson radical of R. As a consequence, we get that _R ω has finite projective dimension if and only if every left (finitely presented) R-module has finite right orthogonal dimension (relative to _R ω). If ω is a tilting module, we then prove that a left R-module has finite right orthogonal dimension (relative to _R ω) if and only if it has a special ω^⊥-preenvelope.     

Injective Envelopes and (Gorenstein) Flat Covers

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of _R R. Then we get that the injective envelope of _R R is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left R-module is (Gorenstein) flat, if and only if the injective envelope of every flat left R-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left R-module is injective, and if and only if the opposite version of one of these conditions is satisfied.     

弱总体维数和模的自同态

本文用模的自同态,给出弱总体维数≤ｎ的环的特征,其中ｎ≥０．设Ｒ为环,部分地回答了下列问题：何时任意有限表现Ｒ－模Ｍ有无穷分解：０→Ｍ→Ｆ₀→Ｆ₁→…→Ｆ_n→…,其中每个Ｆ_i均是有限生成投射的,ｉ＝１,２,…？     

Almost finite modules

Assume that ?(R, m, k) → (S, n, l) is a local homomorphism between commutative noetherian local rings R and S. We say that an S-module M is almost finite over R if it is finitely generated over S (the R-structure on M is induced by ?). We investigate the homological behaviour of such modules, as well as various properties of the rings R and S in the presence of an almost finite module of finite flat dimension over R.     

右IF－环及凝聚环的挠理论

本文研究了右ＩＦ－环的性质，证明出环Ｒ是右ＩＦ－环当且仅当Ｒ是左凝聚环，并且是平坦模；由此证明出右ＩＦ－环与左ＧＱＦ－环是等价的，其次应用右ＩＦ－环研究了凝聚环的挠理论性质，证明出凝聚环与Ｔ－凝聚环的关系。     

Construction of Modules with Finite Homological Dimensions

A new homological dimension, called G^*-dimension, is defined for every finitely generated module M over a local noetherian ring R. It is modeled on the CI-dimension of Avramov, Gasharov, and Peeva and has parallel properties. In particular, a ring R is Gorenstein if and only if every finitely generated R-module has finite G^*-dimension. The G^*-dimension lies between the CI-dimension and the G-dimension of Auslander and Bridger. This relation belongs to a longer sequence of inequalities, where a strict inequality in any place implies equalities to its right and left. Over general local rings, we construct classes of modules that show that a strict inequality can occur at almost every place in the sequence.