Phd records
https://feeds.library.caltech.edu/people/Fisher-James-Louis/Phd.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 19:05:48 +0000Structure theorems for noncommutative complete local rings
https://resolver.caltech.edu/CaltechTHESIS:02222016-135018005
Authors: {'items': [{'id': 'Fisher-James-Louis', 'name': {'family': 'Fisher', 'given': 'James Louis'}}]}
Year: 1969
DOI: 10.7907/BMXA-R647
<p>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)<sup>i</sup> = 0. It is known that if R is complete in the N(R)-adic topology then R is equal to (B)<sub>n</sub>, 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.</p>
<p>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<sub>i</sub>,...,g<sub>k</sub> of B/N(B) over F such that B is a homomorphic image of B/N[[x<sub>1</sub>,…,x<sub>k</sub>;g<sub>1</sub>,…,g<sub>k</sub>]] the power series ring over B/N(B) in noncommuting indeterminates x<sub>i</sub>, where x<sub>i</sub>b = g<sub>i</sub>(b)x<sub>i</sub> for all b ϵ B/N.</p>
<p>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<sub>1</sub>,…,g<sub>k</sub> of a v-ring V such that B is a homomorphic image of V [[x<sub>1</sub>,…,x<sub>k</sub>;g<sub>1</sub>,…,g<sub>k</sub>]].</p>
<p>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. </p>
https://thesis.library.caltech.edu/id/eprint/9577