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Title: Localisations of non-prime orders
Author: Williams, Stephen
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1984
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This thesis is the study of non-prime orders and their localisations It generalises the work of Chamarie, Marubayashi and Fujita on prime Goldie maximal orders and e-vH-orders, as well as moving in new directions. In Chapter two we introduce the notion of an additive regular ring demonstrating their importance in non-prime orders and proving C(A) being an Ore set → C(A) n C(0) = S(A) is an Ore set for A an R-ideal of R. This leads us to consider both the rings Rp and RS(p) (which both coincide on the prime case). In Chapter three we introduce three chain conditions on an e-v H-order namely ꚍNoetherian, r-ꚍ-Noetherian, ꚍ ○- Noetherian and use these to give circumstances for when C(A) or S(A) is an Ore set. In Chapter four we look at localisations of e-vH-orders showing they are vH-orders and discuss the problem of when they have enough v-invertible ideals. In Chapter five we look at the structure of RS(A) for v-invertible ideals A when R has a semi-local quotient ring, and give an intersection theorem analogous to the prime case. In Chapter six we look at the structure of Rp for P a maximal v-invertible ideal and discuss its rank under various chain conditions. In Chapter seven we prove a splitting theorem for ꚍ○-Noetherian e-vH-orders in Artinian quotient rings and give applications. In Chapter eight we look at the structure of e-vH-orders with a finite number of v-idempotent ideals. Finally in Chapter nine examples are given to show the theory is not redundant and further problems are discussed.
Supervisor: Not available Sponsor: Sussex European Research Centre
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics