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Wilkinson, Jeffrey Charles (1983) Skew polynomial rings and overrings. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b3254028~S15
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 ainvariant nilpotent radical.
Finally, A(R, ) is applied to the skew Laurent polynomial ring R[x,x1] 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 ainvariant nilpotent radical.
Finally, A(R, ) is applied to the skew Laurent polynomial ring R[x,x1] 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.
Item Type:  Thesis or Dissertation (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Polynomial rings, Rings (Algebra)  
Official Date:  May 1983  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Hajarnavis, C. R.  
Sponsors:  Commonwealth Scholarship Commission in the United Kingdom  
Format of File:  
Extent:  [5], iii, 182 leaves  
Language:  eng 
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