quasi-Frobenius Rings Research Articles

On semiperfect a -rings

UDC 512.5 A ring is called a right a -ring if every right ideal is automorphism invariant. We describe some properties of a -rings over semiperfect rings. It is shown that an I-finite right a -ring is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if R is an indecomposable (as a ring) I-finite right a -ring not simple with nontrivial idempotents such that every minimal right ideal is a right annihilator and S o c ( R R ) = S o c ( R R ) is essential in R R , then R is a quasi-Frobenius ring and it is also a right q -ring.

Homological objects of min-pure exact sequences

In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are considered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring $R$ is two-sided K\"othe if all right $R$-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with $J(R)^{2}=0$, it is proven that all $R$-modules are min-pure projective if and only if $R$ is either a field or a quasi-Frobenius ring of composition length $2$.

On Annihilators and Quasi-Frobenius Rings

On Annihilators and Quasi-Frobenius Rings

New characterizations of quasi-Frobenius rings

In this paper, we firstly provide several new characterizations of quasi-Frobenius rings by using some generalized injectivity of rings with certain chain conditions. For example, [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is right [Formula: see text], right minfull with ACC on right annihilators; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is two-sided min-CS with ACC on right annihilators in which [Formula: see text]; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is right Johns left [Formula: see text]; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is quasi-dual two-sided [Formula: see text] with ACC on right annihilators. Moreover, it is shown that a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is a left [Formula: see text]-injective left IN-ring with right RMC and [Formula: see text]. Also, we prove that if [Formula: see text] is a right duo, right QF-[Formula: see text] left quasi-duo ring satisfying ACC on right annihilators, then [Formula: see text] is quasi-Frobenius. In this paper, several known results on quasi-Frobenius rings are reproved as corollaries.

Graded (quasi-)Frobenius rings

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.

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.

MODULES AND RINGS IN WHICH EVERY COMPLEMENT IS ISOMORPHIC TO A SUMMAND

In this paper, we introduce a generalization of the well-known CS condition. We say an -module is a - if every complement is isomorphic to a summand. We prove that if is a right CIS-ring right FGF ring, then is a quasi-Frobenius ring, and if is a right CIS-ring right CF ring, then is a right artinian ring. New characterizations of quasi-Frobenius rings are provided by using CIS-rings. Moreover, many of the important propositions related to CS-rings are generalized to CIS-rings also presented.

RD-flatness and RD-injectivity of simple modules

We say that a ring R is a right RDV-ring if each simple right R -module is RD -injective. In this paper, we study the notion of RDV -rings which is a non-trivial generalization of V-rings and Köthe rings. For instance, commutative RD -rings, serial rings and right duo right uniserial rings are RDV -rings. Several characterizations of right RDV -rings are given. Also, it is shown that over a semilocal ring R with Jacobson radical J , each simple right R -module is RD -flat if and only if R is a left RDV -ring, if and only if R ( R / J ) is RD -injective, if and only if ( R / J ) R is RD -flat. As a consequence, we show that a local ring R is a principal ideal ring if and only if R satisfies the ascending chain condition on principal left ideals and R ( R / J ) is RD -injective. In the case of R being either a local left perfect ring or a normal left perfect ring, we have obtained results which state that to check whether every left R -module is RD -injective (or, R is Köthe), it suffices to test only the RD -injectivity of the simple left R -modules. Finally, we give some characterizations of quasi-Frobenius rings by using these concepts.

Coefficient Systems on the Bruhat-Tits Building and Pro-𝑝 Iwahori-Hecke Modules

Let G G be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic p p . Let I I be a pro- p p Iwahori subgroup of G G and let R R be a commutative quasi-Frobenius ring. If H = R [ I ∖ G / I ] H=R[I\backslash G/I] denotes the pro- p p Iwahori-Hecke algebra of G G over R R we clarify the relation between the category of H H -modules and the category of G G -equivariant coefficient systems on the semisimple Bruhat-Tits building of G G . If R R is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler resolution and of the Zelevinski conjecture for smooth G G -representations generated by their I I -invariants. In general, it gives a description of the derived category of H H -modules in terms of smooth G G -representations and yields a functor to generalized ( φ , Γ ) (\varphi ,\Gamma ) -modules extending the constructions of Colmez, Schneider and Vignéras.

On infinite MacWilliams rings and minimal injectivity conditions

We provide a complete answer to the problem of characterizing left Artinian rings which satisfy the (left or right) MacWilliams extension theorem for linear codes, generalizing results of Iovanov [J. Pure Appl. Algebra 220 (2016), pp. 560–576] and Schneider and Zumbrägel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961] and answering the question of Schneider and Zumbragel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961]. We show that they are quasi-Frobenius rings, and are precisely the rings which are a product of a finite Frobenius ring and an infinite quasi-Frobenius ring with no non-trivial finite modules (quotients). For this, we give a more general “minimal test for injectivity” for a left Artinian ring A A : we show that if every injective morphism Σ k → A \Sigma _k\rightarrow A from the k k ’th socle of A A extends to a morphism A → A A\rightarrow A , then the ring is quasi-Frobenius. We also give a general result under which if injective maps N → M N\rightarrow M from submodules N N of a module M M extend to endomorphisms of M M (pseudo-injectivity), then all such morphisms N → M N\rightarrow M extend (quasi-injectivity) and obtain further applications.

MacWilliams extending conditions and quasi-Frobenius rings

MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius thus answering the different questions asked in [13,22]. We also prove that a right perfect, left automorphism-invariant ring is left self-injective. In particular, this yields that if R is a right (or left) artinian, left automorphism-invariant ring, then R is quasi-Frobenius, thus answering a question asked in [13].

GENERALIZATIONS OF C3 MODULES AND C4 MODULES

"Let A be a class of right R-modules that is closed under isomorphisms, and let M be a right R-module. Then M is called A -C3 if, whenever N and K are direct summands of M with N ∩K = 0 and K ∈ A , then N ⊕K is also a direct summand of M; M is called an A -C4 module, if whenever M = A⊕B where A and B are submodules of M and A ∈ A , then every monomorphism f : A → B splits. Some characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple artinian rings, right V-rings, quasi-Frobenius rings and von Neumann regular rings are given."

Unbounded derived categories of small and big modules: Is the natural functor fully faithful?

Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the full subcategory of complexes with finitely generated cohomology modules in two special cases. If $R$ is a quasi-Frobenius ring of infinite global dimension, then this functor is not full. If $R$ has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.

A Note on Co-Harada Rings

In the early 1990s, Harada and Oshiro introduced extending and lifting properties for modules and, simultaneously, considered two new classes of artinian rings which contain quasi-Frobenius (QF-) rings and Nakayama rings: one is the class of right Harada rings and the other is the class of right co-Harada rings. Although QF-rings and Nakayama rings are left-right symmetric, Harada and co-Harada rings are not left-right symmetric. However, Oshiro showed that left Harada rings and right co-Harada rings are coinside. In this paper we provide many characterizations of right co-Harada rings and (right and left) co-Harada rings.

Modules whose t-closed submodules are their homomorphic images

In this paper, we introduce the notion of t-epi-modules. An R-module M is called t-epi if every t-closed submodule of M is a homomorphic image of M. Various properties of these modules are studied. Also, connections between t-epi modules and other related modules are investigated. Finally, quasi-Frobenius rings, right t-semi-simple rings and right S-t-extending rings are characterized in termes of t-epi modules.

When proper cyclics are homomorphic image of injectives

Quasi-Frobenius rings are precisely rings over which any right module is a homomorphic image of an injective module. We investigate the structure of rings whose proper cyclic right modules are homomorphic image of injectives. The class of such rings properly contains that of right self-injective rings. We obtain some structure theorems for rings satisfying the said property and apply them to the Artin algebra case: It follows that an Artin algebra with this property is Quasi-Frobenius.

Some new characterizations of quasi-Frobenius rings by using pure-injectivity

Some new characterizations of quasi-Frobenius rings by using pure-injectivity

Notes on quasi-Frobenius rings

Abstract We give some new characterizations of quasi-Frobenius rings. Namely, we prove that for a ring R, the following statements are equivalent: (1) R is a quasi-Frobenius ring, (2) M 2 ⁢ ( R ) {M_{2}(R)} is right Johns and every closed left ideal of R is cyclic, (3) R is a left 2-simple injective left Kasch ring with ACC on left annihilators, (4) R is a left 2-injective semilocal ring such that R / S l {R/S_{l}} is left Goldie, (5) R is a right YJ-injective right minannihilator ring with ACC on right annihilators.

On C-co-epi-retractable Modules

In this paper, we introduce the notion of c-co-epi-retractable modules. An R-module M is called c-co-epi-retractable if it contains a copy of its factor module by a complement submodule. The ring R is called c-co-pri if RR is c-co-epi-retractable. Conditions are found under which, a c-coepi-retractable module is extending, retractable, semi-simple, quasi-injective, injective and simple. Also, we investigate when c-co-epi-retractable modules have finite uniform dimension. Finally, right SI-rings, semi-simple artinian rings and quasi-Frobenius rings are characterized in termes of c-co-epi-retractable modules.

Quasi-Frobenius Amalgamated Algebras

Let $$f:A\rightarrow B$$ be a homomorphism of commutative rings and let J be an ideal of B. The amalgamation of A with B along J with respect to f is the subring of $$A\times B$$ given by $$A\bowtie ^fJ=\{(a,f(a)+j)\mid a\in A, \, j\in J\}$$ . In this paper, we give some characterizations for the amalgamation construction to be a quasi-Frobenius ring.