A simple characterization of principal ideal domains

Principal Ideal Domains

Last week, Ari taught you about one kind of “simple” (in the nontechnical sense) ring, specifically semisimple rings. These have the property that every module splits as a direct sum of simple modules (in the technical sense). This week, we’ll look at a rather different kind of ring, namely a principal ideal domain, or PID. These rings, like semisimple rings, have the property that every (finit...

Whitehead Modules over Large Principal Ideal Domains

We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.’s of size ≥ א2 have nonfree Whitehead modules even though they are not complete discrete valuation rings. A module M is a Whitehead module if ExtR(M,R) = 0. The second author proved that the problem of whether every Whitehead Z-module is free is independent of ZFC + GCH (cf. [5], ...

(C, A)-Invariance of Modules over Principal Ideal Domains

For discrete-time linear systems over a principal ideal domain three types of (C;A)-invariance can be distinguished. Connections between these notions are investigated. For pure submodules necessary and su cient conditions for dynamic (C;A)-injection invariance are given. Su cient conditions are obtained in the general case. Mathematical Subject Classi cations (1991): 93B07, 93B99, 15A33, 13C99

A Simple Proof of Some Generalized Principal Ideal Theorems

Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans “generalized principal ideal theorem” on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwieciński, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

SAGBI and SAGBI-Gröbner Bases over Principal Ideal Domains