Title:

Skew polynomial rings and overrings

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.
