Prime Ideals and Localization in Noetherian Ore Extensions

Abstract

<p>This thesis studies the prime ideals in a certain class of non-commutative polynomial rings known as Ore extensions. For a right Noetherian Ore extension R[x;σ], the prime ideals are of three types: one type corresponds to the prime ideals of the coefficient ring R, another type corresponds to certain semiprime ideals of R, and the third type is in bijective correspondence with the irreducible polynomials in certain ordinary polynomial rings.</p> <p>The second part of the thesis studies the question of the localizability of prime ideals in Ore extensions of commutative Noetherian rings. It is shown that these rings satisfy Jategaonkar's second layer condition and that the corresponding skew Laurent polynomial ring is Krull symmetric. Using these properties and the classification of prime ideals, a complete description of the obstructions to localizability - the links between prime ideals - is obtained.</p>