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

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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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Polynomial rings, Rings (Algebra)
Official Date:	May 1983
Dates:	Date Event May 1983 UNSPECIFIED
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:	pdf
Extent:	[5], iii, 182 leaves
Language:	eng

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