QF Rings Research Articles

MỘT CHÚ Ý TRÊN VÀNH CS-NỬA ĐƠN

In this paper, we introduce condition (∗) for modules and show that every CS-semisimple ring having condition (∗) is a QF ring.

A Discussion of FGF Conjecture in Some Quasi-Frobenus Rings

A ring R is called right FGF ring, if every finitely generated right R-module embeds in a free right R-module. It is well known that a quasi-Frobenus ring R is right FGF ring, but the converse is still an open question. In this note we give some equivalent additional conditions that convert the right FGF ring to the QF ring. Other known results that characterize the class of quasi-Frobenius rings are fond. In the process, some new results that characterize the class of IF-rings are provided.

C-PURE SUBMODULES AND C-FLAT MODULES

Let R be a ring. A right R-module A is said to be C-flat if the kernel of any epimorphism B → A is C-pure in B, i.e. the induced map Hom(C,B) → Hom(C,A) is surjective for any cyclic right R-module C. Projective modules are C-flat and C-flat modules are weakly-flat and neat-flat. In this article, it is discussed the connections between C-flat, weakly-flat, neat-flat and singly flat modules. It is shown that C-flat modules coincide with singly-projective modules over arbitrary rings. Next, several characterizations of certain classes of rings and modules via C-purity are considered. We prove that, every C-flat module is injective if and only if R is a QF ring. Moreover, we show that R is a CF ring if and only if every FP-injective right R-module is C-flat.

On copure projective and cotorsion modules

Abstract Let R be an IF ring, or be a ring such that each right R-module has a monomorphic flat envelope and the class of flat modules is coresolving. We firstly give a characterization of copure projective and cotorsion modules by lifting and extension diagrams, which implies that the classes of copure projective and cotorsion modules have some balanced properties. Then, a relative right derived functor is introduced to investigate copure projective and cotorsion dimensions of modules. As applications, some new characterizations of QF rings, perfect rings and noetherian rings are given.

On QF rings and artinian principal ideal rings

In this work we give sufficient conditions for a ring $R$ to be quasi-Frobenius, such as $R$ being left artinian and the class of injective cogenerators of $R$-Mod being closed under projective covers. We prove that $R$ is a division ring if and only if $R$ is a domain and the class of left free $R$-modules is closed under injective hulls. We obtain some characterizations of artinian principal ideal rings. We characterize the rings for which left cyclic modules coincide with left cocyclic $R$-modules. Finally, we obtain characterizations of left artinian and left coartinian rings.

A note on generalizations of quasi-Frobenius rings

A ring [Formula: see text] is called quasi-Frobenius, briefly QF, if [Formula: see text] is right (or left) Artinian and right (or left) self-injective. A ring [Formula: see text] is called right co-Harada if every noncosmall right [Formula: see text]-module contains a nonzero projective direct summand and [Formula: see text] satisfies the ACC on right annihilators. The class of co-Harada rings is one of the most interesting generalizations of QF rings. When considering relation between these ring classes, [K. Oshiro, Lifting modules, extending modules and their applications to QF-ring, Hokkaido Math. J. 13 (1984) 310–338, Theorem 4.3] showed that a ring [Formula: see text] is QF if and only if it is right co-Harada ring with [Formula: see text]. In this note, we show that a ring [Formula: see text] is QF if and only if it is right co-Harada ring and satisfies either [Formula: see text] or [Formula: see text].

Cartan-Eilenberg projective, injective and flat complexes

Let R be an associative ring with identity and F a class of R-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg F complexes and extend the basic properties of the class F to the class CE(F). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.

A Note on the Faith–Menal Conjecture

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.

N-Copure projective modules

Let R be a ring, n a fixed nonnegative integer and Fn the class of all left R-modules of flat dimension at most n. A left R-module M is called n-copure projective if ExtR1(M,F) = 0 for any F ∈ Fn. Some examples are given to show that n-copure projective modules need not be m-copure projective whenever m > n. Then we characterize the well-known QF rings and IF rings in terms of n-copure projective modules. Finally, we prove that a ring R is relative left hereditary if and only if every submodule of a projective (or free) left R-module is n-copure projective if and only if idR(N) ≤ 1 for every left R-module N with N ∈ Fn.

On small dual rings

A ring R is called right (small) dual if every (small) right ideal of R is a right annihilator. Left (small) dual rings can be defined similarly. And a ring R is called (small) dual if R is left and right (small) dual. It is proved that R is a dual ring if and only if R is a semilocal and small dual ring. Several known results are generalized and properties of small dual rings are explored. As applications, some characterizations of QF rings are obtained through small dualities of rings.

Homological dimensions of complexes related to cotorsion pairs

Let (A ; B ) be a cotorsion pair in R-Mod. We dene and study notions of A dimension and B dimension of unbounded complexes, which is given by means of dg-projective resolution and dg-injective resolution, respectively. As an application, we extend the Gorenstein at dimension of complexes, which was dened by Iacob. Gorenstein cotorsion, FP-projective, FP-injective, Ding projective, and Ding injective dimension are also extended from modules to complexes. Moreover, we characterize Noetherian rings, von Neumann regular rings, and QF rings by the FP-projective, FP-injective, and Ding projective (injective) dimension of complexes, respectively.

A Construction of Local QF Rings and Its Characterization

We study local artinian (not necessarily commutative) rings of split type and give a construction of those rings. Using our construction, we present various examples of local QF rings of split type and of graded or non-graded type.

Rings whose modules have maximal or minimal projectivity domain

Using the notion of relative projectivity, projective modules may be thought of as being those which are projective relative to all others. In contrast, a module M is said to be projectively poor if it is projective relative only to semisimple modules. We prove that all rings have projectively poor modules. In fact, every ring even has a semisimple projectively poor module. Properties of projectively poor modules are studied and particular emphasis is given to the study of modules over PCI domains; we note that over such domains when all right ideals are principal most modules seem to be either projective or projectively poor. We consider rings over which modules are either projective or projectively poor and call them rings without a p-middle class. We show that a QF ring R with homogeneous right socle and J ( R ) 2 = 0 has no right p-middle class. As we analyze the structure of rings with no right p-middle class, among other results, we show that any such ring is the ring direct sum of a semisimple artinian ring and a ring K which is either zero or an indecomposable ring such that either (i) K is a semiprimary right SI-ring with J ( K ) ≠ 0 , or (ii) K is a semiprimary ring with S o c ( K K ) = Z r ( K ) = J ( K ) ≠ 0 , or (iii) K is a prime ring with S o c ( K K ) = 0 , and either J ( K ) = 0 or J K ( K ) and J ( K ) K are infinitely generated, or (iv) K is a prime right SI-ring with infinitely generated right socle.

Some Results of QF Rings

Let R be a ring. We give some new characterizations of QF under the special annihilators condition. Some known results are obtained as corollaries.

On General Injective Rings with Chain Conditions

Kupisch proved that if R is a left and right artinian QF-2 ring and Sr = Sl, then R is QF. A weaker condition for a ring to be a QF ring was obtained by Dan and Thuyet. They proved that if R is a right artinian QF-2 ring and Sr ≤ Sl, then R is QF. In this paper, we prove that if R is a QF-2 ring satisfying ACC on right annihilators in which Sl ≤ eRR (e.g., Sr ≤ Sl with Sr ≤e RR), then R is QF. It is also proved that R is QF if and only if R is a left ef-extending, right continuous ring with ACC on right annihilators.

Characterizations of QF Rings

A ring R is called “left GP-injective” if for any 0 ≠ a ∈ R, there exists n > 0 such that a n ≠ 0 and a n R = r(l(a n )). It is proved that (1) every right Noetherian left GP-injective ring such that every complement left ideal is a left annihilator is a QF ring, (2) every left GP-injective ring with ACC on left annihilators such that every complement left ideal is a left annihilator is a QF ring, and (3) every left P-injective left CS ring satisfying ACC on essential right ideals is a QF ring. Several well-known results on QF rings are obtained as corollaries.

Rings Whose Modules Have Grade Zero

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if HomR(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith’s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.

On the linear weak topology and dual pairings over rings

In this note we study the weak topology on paired modules over a (not necessarily commutative) ground ring. Over QF rings we are able to recover most of the well-known properties of this topology in the case of commutative base fields. The properties of the linear weak topology and the dense pairings are then used to characterize pairings satisfying the so-called α-condition.

On Nakayama Rings#

Fuller [Fuller, K. R. (1969). On indecomposable injectives over artinian rings. Pacific J. Math. 29:115–135] has proved that a right artinian ring is a right Nakayama and right co-Nakayama ring if and only if it is a two-sided Nakayama artinian ring. Oshiro [Oshiro, K. (1989). On Harada rings I. Math. J. Okayama Univ. 31:161–178] has proved that a right artinian ring is a right Nakayama QF-3 ring if and only if it is a two-sided Nakayama artinian ring. These results poses a question whether a right co-Nakayama ring and right QF-3 ring is two-sided Nakayama artinian. First Theorem 1.4 proves this question affirmatively. Next using this result and properties of one-sided Harada rings, we shall give some characterizations of Nakayama rings.

On H-separable extensions of QF-3 rings

Let a ring A be an H-separable extension of a subring B of A, that is, A ⊗ B A is an A-A-direct summand of a finite direct sum of copies of A. If furthermore, (a) B is a left B-direct summand of A (or (b) A is left B-finitely generated projective), and if B is a right (resp. left) artinian QF-3 ring, then A is also a right (resp. left) artinian QF-3 ring. Consequently, if A is an H-separable extension of a serial ring B with one of the conditions (a), (b), then A is also a serial ring. In particular H-separable extension of a uni-serial ring is always uni-serial.