Overrings 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

SAGBI and SAGBI-Gröbner Bases over Principal Ideal Domains

Binary Quadratic Forms over F [ T ] and Principal Ideal Domains

This paper concerns binary quadratic forms over F[T ]. It develops theory analogous to the theory of binary quadratic forms over Z. Most although not all of the results are almost identical, while some of the proofs require different techniques. In particular, the form class group is determined when the form takes values in a principal ideal domain, and the ideal class group (and class group is...