Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.347716
Title: Skew polynomial rings and overrings
Author: Wilkinson, Jeffrey Charles
ISNI:       0000 0001 3568 4295
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1983
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Abstract:
Given a ring R and a monomorphism :R ->- R , it is possible to construct a minimal overring A(R,Given a ring R and a monomorphism :R ->- R , it is possible to construct a minimal overring A(R,) of R to which a extends as an automorphism - this was done by D.A. Jordan in [163. Chapter 1 presents this construction, and the remainder of the thesis is devoted to the study of the ring A(R,) and its applications. Chapter 2 deals with the ideal structure of A(R,) : the prime, semi prime and nilpotent ideals are examined, and it is shown that if nil left ideals of R are nilpotent, then the nilpotent radical of A(R,) is nil- potent. It is also shown that if R has finite left Goldie dimension n , then the left Goldie dimension of A(R,) cannot exceed n - however, an example is constructed to show that the ascending chain condition on left annihilators need not be passed from R to A(R,) . In chapter 3, several aspects of A(R,) are studied under the assumption that it is left Noetherian, and a question raised by Jordan in[16] is settled by an example where R is a ring of Krull dimension 1 , but A(R,) does not have Krull dimension. Examination of the Jacobson radical of A(R,) , and a proof of the fact that maximal left ideals of left Artinian rings are closed, then leads to a generalization of a result of Jategaonkar, which states that if R is left Artinian, then -1(J(R)) - J(R) • Chapter 4 first finds a condition on R equivalent to A(R,) being a full quotient ring, and then finds a regularity condition on R which is equivalent to A(R, ) having a left Artinian left quotient ring in the case where R is left Noetherian with an a-invariant nilpotent radical. Finally, A(R, ) is applied to the skew Laurent polynomial ring R[x,x-1] where a is a monomorphism, to obtain sufficient conditions for RCx.x'1,«] to be semiprimitive, primitive, and Jacobson. Also, equivalent conditions on R are found for R[x,x -1] to be simple.) of R to which a extends as an automorphism - this was done by D.A. Jordan in [163. Chapter 1 presents this construction, and the remainder of the thesis is devoted to the study of the ring A(R,) and its applications. Chapter 2 deals with the ideal structure of A(R,) : the prime, semi prime and nilpotent ideals are examined, and it is shown that if nil left ideals of R are nilpotent, then the nilpotent radical of A(R,) is nil- potent. It is also shown that if R has finite left Goldie dimension n , then the left Goldie dimension of A(R,) cannot exceed n - however, an example is constructed to show that the ascending chain condition on left annihilators need not be passed from R to A(R,) . In chapter 3, several aspects of A(R,) are studied under the assumption that it is left Noetherian, and a question raised by Jordan in[16] is settled by an example where R is a ring of Krull dimension 1 , but A(R,) does not have Krull dimension. Examination of the Jacobson radical of A(R,) , and a proof of the fact that maximal left ideals of left Artinian rings are closed, then leads to a generalization of a result of Jategaonkar, which states that if R is left Artinian, then -1(J(R)) - J(R) • Chapter 4 first finds a condition on R equivalent to A(R,) being a full quotient ring, and then finds a regularity condition on R which is equivalent to A(R, ) having a left Artinian left quotient ring in the case where R is left Noetherian with an a-invariant nilpotent radical. Finally, A(R, ) is applied to the skew Laurent polynomial ring R[x,x-1] where a is a monomorphism, to obtain sufficient conditions for RCx.x'1,«] to be semiprimitive, primitive, and Jacobson. Also, equivalent conditions on R are found for R[x,x -1] to be simple.
Supervisor: Not available Sponsor: Commonwealth Scholarship Commission
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.347716  DOI: Not available
Keywords: QA Mathematics
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