Balanced local rings with commutative residue fields

Balanced Local Rings with Commutative Residue Fields by Vlastimil Dlab and Claus

1. Let R be a ring with unity. An R-module M is said to be balanced or to have the double centralizer property, if the natural homomorphism from R to the double centralizer of M is surjective. If all left and right K-modules are balanced, R is called balanced. It is well known that every artinian uniserial ring is balanced. In [5], J. P. Jans conjectured that those were the only (artinian) bala...

Commutative Rings with Finite Quotient Fields

We consider the class of all commutative reduced rings for which there exists a finite subset T ⊂ A such that all projections on quotients by prime ideals of A are surjective when restricted to T . A complete structure theorem is given for this class of rings, and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals. Introduction Our aim i...

Commutative Rings and Finite Fields

Invertible Modules for Commutative S-algebras with Residue Fields

The aim of this note is to understand invertible modules over a commutative Salgebra in the sense of Elmendorf, Kriz, Mandell & May in some very well-behaved cases. Our main result shows that as long as the commutative S-algebra R has ‘reductions mod m’ for all maximal ideals m ⊳ R∗, and Noetherian localisations (R∗)m , then for every invertible R-module U , U∗ = π∗U is an invertible R∗-module.

Residue Fields of Zero-Dimensional Rings

We begin by deening a couple of terms. If F = fF i g i2I is a family of elds, we say that F is the family of residue elds of the ring T if there exists a bijection g : I ! MaxSpec(T) such that F i ' T=g(i) for each i 2 I. Note that the deenition takes into account the multiplicity with which a given eld occurs in F. We say that F is realizable if F is the family of residue elds of a zero-dimens...