quasi-Frobenius Rings Research Articles

Quasi-Frobenius X-Rings

In a recent study of a specific class of quasi-Frobenius rings, Feller has found it useful to introduce the X-rings ([3]). He suggested among others the following topics:(A)Determine the properties of completely indecomposable rings and matrix rings over completely indecomposable rings.(B)Determine the properties of modules over quasi-Frobenius X-rings.We point out that the completely indecomposable rings are the local quasi-Frobenius rings. Problems (A) and (B) then lead naturally to semi-local quasi-Frobenius rings, and to matrix algebra over local quasi-Frobenius rings. These types of rings are discussed in sections 1 and 2.

Two characterizations of quasi-Frobenius rings

Two characterizations of quasi-Frobenius rings

A Note on Quasi-Frobenius Rings and Ring Epimorphisms

In this note, we characterize quasi-Frobenius rings by a weakened form of the usual condition, that every ideal is an annihilator ideal.We then apply this result to pure rings in the sense of Cohn and to dominant rings, a concept arising in the study of ring epimorphisms. All rings considered have a unit element.

A Generalization of Quasi-Frobenius Rings

A Generalization of Quasi-Frobenius Rings

A generalization of quasi-Frobenius rings

As a generalization of quasi-Frobenius rings, G. Azumaya [2] has investigated a ring with the property that every faithful left module is a generator' in the category of left mnodules, and he has proved that such a ring is left self-injective and a direct sum of indecomposable left ideals having minimal left ideals. In his proof, however, the existence of the faithful injective module plays an important role, while the injective module is not necessarily finitely generated,2 even if the ring is a left Artinian ring. Therefore, there naturally arises a problem: Is an Artinian ring quasi-Frobenius if every finitely generated faithful module is a generator? The purpose of this paper is to give an affirmative answer to this problem as a direct consequence from a more general result which is rather similar to that of Azumaya stated above and related to perfect rings introduced by H. Bass [3]. Throughout this paper we shall assume that the ring R has identity element 1 and all modules over it are unital. For a subset A of R we shall denote

Some remarks on quasi-Frobenius rings

Some remarks on quasi-Frobenius rings

A note on quasi-Frobenius rings

A note on quasi-Frobenius rings

On orders in quasi-Frobenius rings

On orders in quasi-Frobenius rings

A Type of Quasi-Frobenius Ring

In [3], the author proved that a ring R with identity is right noetherian and right injective if and only if R is a direct sum of a finite number of uniform right ideals, which are completely primary in the sense of that paper. In this paper, we shall determine the structure of such rings in the case where the sum of the isomorphic uniform components are twosided ideals. The ring is found to be a direct sum of total matrix rings over local rings.

A generalization of quasi-frobenius rings

A generalization of quasi-frobenius rings

Rings with Ascending Condition on Annihilators

Quasi-frobenius (= QF) rings have many interesting characterizations. One such, due to Ikeda [17] is that these rings are right (left) artinian and right (left) self-injective. Thus, if R is QF, then R is right (left) noetherian, so each direct sum of injective right R-modules is injective: in particular, each free, hence, each projective, R-module is injective. One object of this paper is to report that this property characterizes QF-rings:

Bicontinuous Isomorphisms between Two Closed Left Ideals of a Compact Dual Ring

A quasi-Frobenius ring is a ring with minimum condition satisfying the conditions r(l)H)) = H and l(r(L)) = L for right ideals H and left ideals L where r(S) (l(S)) denotes the right (left) annihilator of a subset S of the ring. Nakayama first defined and studied such rings (8; 9) and they have been studied by a number of authors (2; 3; 4; 6). A dual ring is a topological ring satisfying the conditions r(l)H)) = H and l(r)H)) = L for closed right ideals H and closed left ideals L. Baer (1) and Kaplansky (7) introduced the notion of such rings, which is a natural generalization of that of quaso-Frobenius rings. Numakura studied the analogy between dual rings and quasi-Frobenius rings in (10).

A remark concerning quasi-Frobenius rings

A remark concerning quasi-Frobenius rings

Remarks concerning two quasi-Frobenius rings with isomorphic radicals

Remarks concerning two quasi-Frobenius rings with isomorphic radicals

Note on some generalizations of quasi-Frobenius rings

Note on some generalizations of quasi-Frobenius rings

Quasi-Frobenius rings and Galois theory

Quasi-Frobenius rings and Galois theory

On the Dimension of Modules and Algebras, II: (Frobenius Algebras and Quasi-Frobenius Rings)

In this paper we study Frobenius algebras and quasi-Frobenius rings with particular emphasis on their cohomological dimensions. For definitions of these cohomological dimensions we refer the reader to Cartan-Eilenberg [3] or Eilenberg [4].

On some characteristic properties of quasi-Frobenius and regular rings

Recently the first writer [1] gave a characterization of quasiFrobenius rings, introduced formerly by the second writer [3], in terms of a condition proposed by K. Shoda, which reads: A ring A satisfying minimum condition and possessing a unit element is a quasi-Frobenius ring if and only if A satisfies the following condition :1 (a) every (A-left-) homomorphism of a left-ideal of A into A may be given by the right multiplication of an element of A. In the present note we shall offer a simpler2 proof of this, making use of the second writer's former characterization of quasi-Frobenius rings and a theorem in a previous joint note of the writers. The present approach starts, contrary to the one in [1 ],3 with a theorem (Theorem 1) which is independent of any chain condition (or is concerned with maximum condition at most (corollary to Theorem 1) and which is perhaps of interest by itself. We shall also show that a remark at the end of [1], concerning semisimple rings with minimum condition, may be freed from chain condition, to yield a certain characterization of von Neumann's regular rings (Theorem 3).

Shoda's condition on quasi-frobenius rings

Shoda's condition on quasi-frobenius rings

A characterization of quasi-Frobenius rings

A characterization of quasi-Frobenius rings