Carl Faith

Lectures on injective modules and quotient rings

Injective modules.- Essential extensions and the injective hull.- Quasi-Injective modules.- Radical and semiprimitivity in rings.- The endomorphism ring of a quasi-injective module.- Noetherian, artinian, and semisimple modules and rings.- Rational extensions and lattices of closed submodules.- Maximal quotient rings.- Semiprime rings with maximum condition.- Nil and singular ideals under maximum conditions.- Structure of noetherian prime rings.- Maximal quotient rings.- Quotient rings and direct products of full linear rings.- Johnson rings.- Open problems.

Rings and things and a fine array of twentieth century associative algebra

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: Schurs 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 Maschkes 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 Dedekinds 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.

On Noetherian prime rings

Classical left quotient rings are defined symmetrically. R is right (resp. left) quotient-simple in case R has a classical right (resp. left) quotient ring S which is isomorphic to a complete ring Dn of n X n matrices over a (not necessarily commutative) field D. R is quotient-simple if R is both left and right quotient-simple. Goldie [2] has determined that a ring R is right quotient-simple if and only if R is a prime ring satisfying the maximum conditions on complement and annihilator right ideals. In particular, any right noetherian prime ring is right quotient-simple. (See also Lesieur-Croisot [1 ] .) A (not necessarily commutative) integral domain K is a right Ore domain in case K possesses a classical right quotient field K. Observe that if K is a right Ore domain, then, for each natural number n, the ring Kn of all n X n matrices over K is right quotient-simple, and (K)n is its classical right quotient ring. A consequence of our main result (Theorem 2.3) is that the right quotientsimple rings can be determined as the class of intermediate rings of the extensions (K)n over Kn, n ranging over all natural numbers, and K ranging over all right Ore domains. Theorem 2.3 is much more precise. As a corollary we rederive a theorem of Goldie [3] on principal right ideal prime rings.

Associated primes in commutative polynomial rings

The aim of this paper is to give a new and direct proof of the theorem.

A correspondence theorem for projective modules and the structure of simple Noetherian rings

Much of Chapter 4 is devoted to the exposition of the structure theory of simple right Noetherian rings. The basic tools generalize the theorems of Morita [58] characterizing similarity of two rings A and B, that is, when there is an equivalence mod-A ≈ mod-B of categories. Morita’s characterization 4.29 predicates the existence of a finitely generated projective module P which is a generator of the category mod-B such that A is isomorphic to End B P. In the Morita situation 4.30 there is a lattice isomorphism

INDECOMPOSABLE INJECTIVE MODULES AND A THEOREM OF KAPLANSKY

\left\{ {\begin{array}{*{20}{c}} {right A - submodules of \to right ideals of B} \\ {IP \leftrightarrow I} \end{array}} \right.

Semiperfect Prüfer rings and FPF rings

sending (B, A)-submodules onto ideals of B. More generally, 4.7, if U is any finitely generated projective faithful left B-module over any ring B, and A = End B U then there is a lattice isomorphism