A ring is called quasi-Frobenius if it is right or left selfinjective, and right or left artinian (all four combinations are equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field, and the subject is intimately related to duality, the duality from right to left modules induced by the hom functor and the duality related to annihilators. The present extent of the theory is vast, and this book makes no attempt to be encyclopedic; instead it provides an elementary, self-contained account of the basic facts about these rings at a level allowing researchers and graduate students to gain entry to the field.

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Quasi-Frobenius Rings

Part I An array of twentieth century associative algebra: Direct product and sums of rings and modules and the structure of fields Introduction to ring theory: Schur's lemma and semisimple rings, prime and primitive rings, nil, prime and Jacobson radicals Direct sum decompositions of projective and injective modules Direct product decompositions of von Neumann regular rings and self-injective rings Direct sums of cyclic modules When injectives are flat: Coherent FP-injective rings Direct decompositions and dual generalizations of Noetherian rings Completely decomposable modules and the Krull-Schmidt-Azumaya theorem Polynomial rings over Vamosian and Kerr rings, valuation rings and Prufer rings Isomorphic polynomial rings and matrix rings Group rings and Maschke's theorem revisited Maximal quotient rings Morita duality and dual rings Krull and global dimensions Polynomial identities and PI-rings Unions of primes, prime avoidance, associated prime ideals, ACC on irreducible ideals and annihilator ideals in commutative rings Dedekind's theorem on the independence of automorphisms revisited Part II Snapshots of some mathematical friends and places: Snapshots of some mathematical friends and places Index to part II (snapshots) Bibliography Register of names Index of terms and authors of theorems

/pdf/rings-and-things-and-a-fine-array-of-twentieth-century-4feqme3n14.pdf

Rings and Things and a Fine Array of Twentieth Century Associative Algebra

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m, n)-injective if every right R-homomorphism from an n-generated submodule of Rm to M extends to one from Rm to M. This definition unifies several definitions on generalizations of injectivity of modules. The aim of this paper is to investigate properties of the (m, n)-injective modules. Various results are developed, many extending known results.

ON (m, n)-INJECTIVITY OF MODULES

Embedding modules in projectives: A report on a problem

In this paper we continue our study of right Johns rings, that is, right Noetherian rings in which every right ideal is an annihilator. Specifically we study strongly right Johns rings, or rings such that every n x n matrix ring Rn is right Johns. The main theorem (Theorem 1.1) characterizes them as the left FP-injective right Noetherian rings, a result that shows that not all Johns rings are strong. (This first was observed by Rutter for Artinian Johns rings; see Theorem 1.2.) Another characterization is that all finitely generated right R-modules are Noetherian and torsionless, that is, embedded in a product of copies of R . A corollary to this is that a strongly right Johns ring R is preserved by any group ring RG of a finite group (Theorem 2.1). A strongly right Johns ring is right FPF (Theorem 4.2).

/pdf/the-structure-of-johns-rings-4ac0yqa3mk.pdf

The structure of Johns rings

Annihilator conditions in Noetherian rings

Chain conditions and nil ideals

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Annihilator conditions in Noetherian rings

Chain conditions and nil ideals