Citation
Fisher, James Louis (1969) Structure Theorems for Noncommutative Complete Local Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BMXAR647. https://resolver.caltech.edu/CaltechTHESIS:02222016135018005
Abstract
If R is a ring with identity, let N(R) denote the Jacobson radical of R. R is local if R/N(R) is an artinian simple ring and ∩N(R)^{i} = 0. It is known that if R is complete in the N(R)adic topology then R is equal to (B)_{n}, the full n by n matrix ring over B where E/N(E) is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following.
If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms g_{i},...,g_{k} of B/N(B) over F such that B is a homomorphic image of B/N[[x_{1},…,x_{k};g_{1},…,g_{k}]] the power series ring over B/N(B) in noncommuting indeterminates x_{i}, where x_{i}b = g_{i}(b)x_{i} for all b ϵ B/N.
Another theorem generalizes this result to complete local rings which have suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N(B) a finite field. If N(B) is finitely generated as a left ideal by k elements then there exist automorphisms g_{1},…,g_{k} of a vring V such that B is a homomorphic image of V [[x_{1},…,x_{k};g_{1},…,g_{k}]].
In both these results it is essential to know the structure of N(B) as a two sided module over a suitable subring of B.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  (Mathematics)  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  17 March 1969  
Funders: 
 
Record Number:  CaltechTHESIS:02222016135018005  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:02222016135018005  
DOI:  10.7907/BMXAR647  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9577  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  23 Feb 2016 15:52  
Last Modified:  26 Apr 2024 23:36 
Thesis Files

PDF
 Final Version
See Usage Policy. 1MB 
Repository Staff Only: item control page