On Frobeniusean Algebras. I

Abstract. Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterexamples to several naturally raised situations in the process.2000 Mathematics Subject Classification. 16D25, 16D40, 16E50.

show abstract

“…A quasi-Frobenius ring, introduced by Nakayama in 1939 [28], is defined to be right Artinian and right self-injective. A ring R is quasi- Proof.…”

Section: Corollary 313 [37 Theorem 4] a Ring R Is Strongly Regularmentioning

confidence: 99%

On Rings Whose Right Annihilators Are Bounded

2009

View full text Add to dashboard Cite

show abstract

“…In such a case, the concept of left QF-3 and right QF-3 coincide.The study of QF-3 rings and algebras and many other such classes of rings had its origin in the now classic papers of Nakayama [10,11]. He was an outstanding pioneer in algebra for many years, and we acknowledge our great debt to him and to his many excellent papers.…”

mentioning

confidence: 95%

A Characterization of QF-3 Rings

Wu¹,

Mochizuki

Jans

1966

Nagoya Mathematical Journal

View full text Add to dashboard Cite

To the memory of TADASI NAKAYAMA A left QF-3 ring R is one in which R R> the ring considered as a left module over itself, can be embedded in a projective infective left R module Q( R R).QF-3 rings were introduced by Thrall [14] and have been studied and characterized by a number of authors [5,8,9,12,13, 15] usually restricted to the case of algebras over a field. In such a case, the concept of left QF-3 and right QF-3 coincide.The study of QF-3 rings and algebras and many other such classes of rings had its origin in the now classic papers of Nakayama [10,11]. He was an outstanding pioneer in algebra for many years, and we acknowledge our great debt to him and to his many excellent papers.In this note we shall restrict consideration to the class of rings with minimum condition in left ideals and within the class we shall give a new characterization of left QF-3 rings. Actually, our characterization holds for a more general class of rings. See the remarks at the end of the note.Incidently, for rings with minimum codition both on left and on right ideals, we do not know if the concepts of left QF-3 and right QF-3 coincide as they do for the case of finite dimensional algebras over fields.

show abstract

“…An algebra A is called selfinjective if A ∼ = D(A) in mod A, that is, the projective A-modules are injective. By a classical result due to Nakayama [30], a basic algebra A is selfinjective if and only if A is a Frobenius algebra, that is, there exists a nondegenerate K-bilinear form (−, −) : A × A → K satisfying the associativity condition (ab, c) = (a, bc) for all elements a, b, c ∈ A. Moreover, an algebra A is said to be symmetric if A and D(A) are isomorphic as A-A-bimodules, or equivalently, there exists an associative nondegenerate symmetric K-bilinear form (−, −) : A × A → K. An important class of selfinjective algebras is formed by the orbit algebrasB/G, whereB is the repetitive algebra (locally finite-dimensional, without identity) , which is a symmetric algebra.…”

mentioning

confidence: 99%