Noetherian $\mathrm{QF}-3$ rings and two-sided quasi-Frobenius maximal quotient rings

The Quest for Quotient Rings (Of Noncommutative Noetherian Rings)

Articles on the history of mathematics can be written from many different perspectives. Some aim to survey a more or less wide landscape, and require the observer to watch from afar as theories develop and movements are born or become obsolete. At the other extreme, there are those that try to shed light on the history of particular theorems and on the people who created them. This article belo...

Fp-injective and Weakly Quasi-frobenius Rings

The classes of FP -injective and weakly quasi-Frobenius rings are investigated. The properties for both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we characterize the classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G) is FP -injective (weakly quasi-Frobenius...

A Note on Quasi-Frobenius Rings

The Faith-Menal conjecture says that every strongly right Johns ring is QF . The conjecture is also equivalent to say every right noetherian left FP -injective ring is QF . In this short article, we show that the conjecture is true under the condition( a proper generalization of left CS condition) that every nonzero complement left ideal is not small( a left ideal I is called small if for every...

Artinian and Noetherian Rings

Fully Bounded Noetherian Rings

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover resul...