Algebraic properties of twisted polynomial rings

Abstract

In this thesis, we study the twisted polynomial rings and determine several of their intrinsic properties. -- In chapter I, we define the twisted polynomial ring Rα,ψ[X], prove it is a ring and establish some preliminary results. -- In chapter II, we study the chain conditions; here the Hilbert basis theorem is extended to twisted polynomial rings. -- In chapter III, we are concerned with the Noether radical of twisted polynomial rings with zero derivation. -- In chapter lV, we describe the endomorphisms of a twisted polynomial ring which leave the coefficients unchanged. -- In chapter V, we consider only twisted polynomial rings with zero derivation. Here we study the automorphisms which restrict to the identity-map on the ring of coefficients. Complete results are obtained for commutative rings with a regular element. – If two twisted polynomial rings are isomorphic what can be said, about the rings of coefficiants? We look into this question in chapter VI and, in chapter VII, we study in detail the example given by M. Hochster showing that not every ring is invariant.