Polynomial rings and their projective modules

Projective modules over polynomial rings and dynamical Gröbner bases

Modules over Differential Polynomial Rings

This note announces a number of results on the structure of differential modules over differential rings, where differential ring means a ring with a family of derivations and differential module means a module having a family of operators compatible with the derivations of the ring. To fix notation, throughout the paper we let A denote an associative ring, M = AM an 4-module, k the correspondi...

On Projective Modules over Semi-hereditary Rings

This theorem, already known for finitely generated projective modules[l, I, Proposition 6.1], has been recently proved for arbitrary projective modules over commutative semi-hereditary rings by I. Kaplansky [2], who raised the problem of extending it to the noncommutative case. We recall two results due to Kaplansky: Any projective module (over an arbitrary ring) is a direct sum of countably ge...

The Euler class groups of polynomial rings and unimodular elements in projective modules

Let A be a commutative noetherian ring of dimension n. Let P be a projective A[T ]-module. Plumstead ([P]) proved that if rankP > dimA then P splits off a free summand of rank one. It is natural to ask what happens when rankP = dimA. In this paper we investigate this question when P has trivial determinant. Let α : P I be a generic surjection (i.e. I ⊂ A[T ] is an ideal of height n). It is prov...

Codes as Modules over Skew Polynomial Rings

In previous works we considered codes defined as ideals of quotients of non commutative polynomial rings, so called Ore rings of automorphism type. In this paper we consider codes defined as modules over non commutative polynomial rings, removing therefore some of the constraints on the length of the codes defined as ideals. The notion of BCH codes can be extended to this new approach and the c...