quasi-Frobenius Research Articles

On Automorphism-Invariant Rings with Chain Conditions

It is shown that if R is a right automorphism-invariant ring and satisfies ACC on right annihilators, then R is a semiprimary ring. By this useful fact, we study finiteness conditions which ensure an automorphism-invariant ring is quasi-Frobenius (QF). Thus, we prove, among other results, that: (1) R is QF if and only if R is right automorphism-invariant, right min-CS and satisfies ACC on right annihilators; (2) R is QF if and only if R is left Noetherian, right automorphism-invariant and every complement right ideal of R is a right annihilator; (3) If R is right CPA, right automorphism-invariant and every complement right ideal of R is a right annihilator, then R is QF.

Pseudo-Frobenius Graded Algebras with Enough Idempotents

We introduce and study the notion of pseudo-Frobenius graded algebra with enough idempotents, showing that it follows the pattern of the classical concept of pseudo-Frobenius (PF) and quasi-Frobenius (QF) ring, in particular finite dimensional self-injective algebras, as studied by Nakayama, Morita, Faith, Tachikawa, etc. We show that such an algebra is characterized by the existence of a graded Nakayama form. Moreover, we prove that the pseudo-Frobenius property is preserved and reflected by covering functors, a fact which makes the concept useful in representation theory.

$\mathfrak{g}$-quasi-Frobenius Lie algebras

A Lie version of Turaev’s $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $\mathfrak{g}$-quasi-Frobenius Lie algebra for $\mathfrak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\mathfrak{q},\beta )$ together with a left $\mathfrak{g}$-module structure which acts on $\mathfrak{q}$ via derivations and for which $\beta $ is $\mathfrak{g}$-invariant. Geometrically, $\mathfrak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group $G$ which acts via symplectic Lie group automorphisms. In addition to geometry, $\mathfrak{g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\mathfrak{g}$-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $\mathbf{Rep}(\mathfrak{g})$. If $\mathfrak{g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in [16] suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf{Rep}(D(\mathfrak{g}))$, where $D(\mathfrak{g})$ is the associated (semiclassical) Drinfeld double. We show that if $\mathfrak{g}$ is a quasitriangular Lie bialgebra, then every $\mathfrak{g}$-quasi-Frobenius Lie algebra has an induced $D(\mathfrak{g})$-action which gives it the structure of a $D(\mathfrak{g})$-quasi-Frobenius Lie algebra.

A fuzzy characterization of QF rings

Abstract Let R be a ring. R is called a quasi-Frobenius (QF) ring if R is right artinian and R R is an injective right R-module. In this article, we introduce (weak) fuzzy homomorphisms of modules to obtain a fuzzy characterization of QF rings. We also obtain some fuzzy characterizations of right artinian rings and right CF rings. These results throw new light on the research of QF rings and the related CF conjecture. MSC:03E72, 16L60.

THE FGF CONJECTURE AND THE SINGULAR IDEAL OF A RING

We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its left singular ideal. We deduce that a left FGF is Quasi-Frobenius (QF) under this assumption. This clarifies the role played by the Jacobson radical and the singular left ideal in the FGF and CF conjectures. In Sec. 3 of the paper, we study the structure of left artinian left CF rings. We prove that they are left continuous and left CEP rings.

FACTOR RINGS OF PSEUDO-FROBENIUS RINGS

If R is right pseudo-Frobenius (= PF), and A is an ideal, when is R/A right PF? Our main result, Theorem 3.7, states that this happens iff the ideal A′ of the basic ring B of R corresponding to A has left annihilator F in B generated by a single element on both sides. Moreover, in this case B/A′ ≈ F in mod-B, (see Theorem 3.5), a property that does not extend to R, that is, in general R/A is not isomorphic to the left annihilator of A. (See Example 4.3(2) and Theorem 4.5.) Theorem 4.6 characterizes Frobenius rings among quasi-Frobenius (QF) rings. As an application of the main theorem, in Theorem 3.9 we prove that if A is generated as a right or left ideal by an idempotent e, then e is central (and R/A is then trivially right PF along with R). This generalizes the result of F. W. Anderson for quasi-Frobenius rings. (See Theorem 2.2 for a new proof.). In Proposition 1.6, we prove that a generalization of this result holds for finite products R of full matrix rings over local rings; namely, an ideal A is finitely generated as a right or left ideal iff A is generated by a central idempotent. We also note a theorem going back to Nakayama, Goursaud, and the author that every factor ring of R is right PF iff R is a uniserial ring. (See Theorem 5.1.).

Primeness Conditions on the Lattice of Hereditary Pretorsion Classes#

A lattice ordered monoid is a structure ⟨L;⊕, 0 L ; ≤⟩ where ⟨L;⊕, 0 L ⟩ is a monoid, ⟨L; ≤ ⟩ is a lattice and the binary operation ⊕ distributes over finite meets. If R is an arbitrary ring with identity, then the set ptors R of all hereditary pretorsion classes in the category of right R-modules is an example of a lattice ordered monoid. It is known that right strongly prime and right strongly semiprime rings are characterizable by means of first order sentences in the language of ptors R. A notion of weak primeness and weak semiprimeness is introduced. These are defined by means of first order sentences in ptors R. It is shown that in the presence of a weak commutativity condition, which is also defined in the language of ptors R, the notions of weak primeness [resp. weak semiprimeness] and strong primeness [resp. strong semiprimeness] coincide. It is also proved that within the class of commutative rings, Quasi–Frobenius rings are characterizable by means of a simple sentence in ptors R.

Linear Recurring Arrays, Linear Systems and Multidimensional Cyclic Codes over Quasi-Frobenius Rings

This paper generalizes the duality between polynomial modules and their inverse systems (Macaulay), behaviors (Willems) or zero sets of arrays or multi-sequences from the known case of base fields to that of commutative quasi-Frobenius (QF) base rings or even to QF-modules over arbitrary commutative Artinian rings. The latter generalization was inspired by the work of Nechaev et al. who studied linear recurring arrays over QF-rings and modules. Such a duality can be and has been suggestively interpreted as a Nullstellensatz for polynomial ideals or modules. We also give an algorithmic characterization of principal systems. We use these results to define and characterize n-dimensional cyclic codes and their dual codes over QF rings for n>1. If the base ring is an Artinian principal ideal ring and hence QF, we give a sufficient condition on the codeword lengths so that each such code is generated by just one codeword. Our result is the n-dimensional extension of the results by Calderbank and Sloane, Kanwar and Lopez-Permouth, Z. X. Wan, and Norton and Salagean for n=1.

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ⊃ R 2 ⊃ J ⊃ Rn (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R 2 such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = Gk = (2) × ⃛ × (2). Commutative Gorenstein rings appeared at first in the paper [3]. Torsion-free modules over commutative Gorenstein domains were investigated in [1]. Noncommutative Gorenstein orders were considered in [2] and [10]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5]. Arbitrary tiled orders were considered in [4], 11-14.

Rings whose cyclics are essentially embeddable in projective modules

A ring R is quasi-Frobenius (QF) if and only if every right R-module is embeddable in a projective module. We call a ring R a right (left) CEPring if each cyclic right (left) R-module is essentially embeddable in a projective module. Examples of right CEP-rings include QF-rings and right uniserial rings. Indeed R is a QF-ring if and only if R is both a right and left CEP-ring [S]. A right CEP-ring which is QF-3 is shown to be QF (Theorem 3.3). Semiperfect CEP-rings and rings, each of whose homomorphic images is a right CEP-ring, are characterized in Theorems 5.2 and 6.2, respectively. The last section deals with split extensions of right uniserial rings as examples of CEP-rings.

Group Actions on Q-F-Rings

Let B be a ring, G a finite group of automorphisms acting on B and ${B^G}$ the fixed subring of B. We give an example of a B which is quasi-Frobenius (Q-F) such that ${B^G}$ is not quasi-Frobenius.

Group actions on Q-F-rings

Let B be a ring, G a finite group of automorphisms acting on B and B G {B^G} the fixed subring of B. We give an example of a B which is quasi-Frobenius (Q-F) such that B G {B^G} is not quasi-Frobenius.

Some generalization of quasi-Frobenius algebras

Introduction. Let 21 be an algebra over a field f. 21 is called quasiFrobenius (QF) if it has a unit element and if every primitive right ideal is dual to a primitive left ideal; or, equivalently, if 21 has a unit element and if every indecomposable direct constituent of the right regular representation is equivalent to an indecomposable direct constituent of the left regular representation. Properties of QF algebras and rings have been treated by Nakayama, Nesbitt, and the author [2, 3, 5, 6](1). Some of the most important properties of QF algebras do not characterize these algebras, but occur in more extensive classes. This leads us to the definitions that follow. Throughout this paper 21 is a f-algebra with unity element. QF-1 Algebra. { is said to be a QF-1 algebra if every faithful representation of 21 is its own second commutator. QF-2 Algebra. 21 is said to be a QF-2 algebra if every primitive left ideal, and every primitive right ideal, of 21 has a unique minimal subideal. A faithful representation Q3 of an algebra 21 is said to be a minimal faithful representation if deletion of any direct constituent of Q3 leaves a nonfaithful representation, that is, if the corresponding space V is the direct sum of VI and V2 with V20 then Q31 is not faithful. QF-3 Algebra. 21 is said to be a QF-3 algebra if it has a unique minimal faithful representation. We shall use the notation QF-12 to describe an algebra which is both QF-1 and QF-2, and so on. Every QF algebra is QF-123. It is the purpose of the present paper to initiate the study of the above classes of algebras. ?1 contains definitions and notations for the paper. ?2 gives an example to show that the class QF-1 is more general than the class QF. ?3 contains a theorem which gives an equivalent definition for QF-2 algebras, ??4 and 5 discuss conditions under which a QF-2 algebra is QF, and give some properties of the Cartan invariants of QF-2 algebras. ?6 contains the proof that every QF-2 algebra is also a QF-3 algebra. ??7 and 8 treat QF-3 algebras and include examples which illustrate the essentially more general character of QF-3 algebras as compared with QF-2 algebras. ?9 treats a necessary condition on the class QF-13. Some of the above definitions are given in the language of ideal theory