Left 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], ...

Lexicodes over Finite Principal Left Ideal Rings

Let R be a finite principal left ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module R is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combination with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull o...

(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

SAGBI and SAGBI-Gröbner Bases over Principal Ideal Domains