Citation
Gordon, Robert (1966) Rings faithfully represented on their left socle. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:09252015104533327
Abstract
In 1964 A. W. Goldie [1] posed the problem of determining all rings with identity and minimal condition on left ideals which are faithfully represented on the right side of their left socle. Goldie showed that such a ring which is indecomposable and in which the left and right principal indecomposable ideals have, respectively, unique left and unique right composition series is a complete blocked triangular matrix ring over a skewfield. The general problem suggested above is very difficult. We obtain results under certain natural restrictions which are much weaker than the restrictive assumptions made by Goldie.
We characterize those rings in which the principal indecomposable left ideals each contain a unique minimal left ideal (Theorem (4.2)). It is sufficient to handle indecomposable rings (Lemma (1.4)). Such a ring is also a blocked triangular matrix ring. There exist r positive integers K_{1},..., K_{r} such that the i,j^{th} block of a typical matrix is a K_{i} x K_{j} matrix with arbitrary entries in a subgroup D_{ij} of the additive group of a fixed skewfield D. Each D_{ii} is a subskewfield of D and D_{ri} = D for all i. Conversely, every matrix ring which has this form is indecomposable, faithfully represented on the right side of its left socle, and possesses the property that every principal indecomposable left ideal contains a unique minimal left ideal.
The principal indecomposable left ideals may have unique composition series even though the ring does not have minimal condition on right ideals. We characterize this situation by defining a partial ordering ρ on {i, 2,...,r} where we set iρj if D_{ij} ≠ 0. Every principal indecomposable left ideal has a unique composition series if and only if the diagram of ρ is an inverted tree and every D_{ij} is a onedimensional left vector space over D_{ii} (Theorem (5.4)).
We show (Theorem (2.2)) that every ring A of the type we are studying is a unique subdirect sum of less complex rings A_{1},...,A_{s} of the same type. Namely, each A_{i} has only one isomorphism class of minimal left ideals and the minimal left ideals of different A_{i} are nonisomorphic as left Amodules. We give (Theorem (2.1)) necessary and sufficient conditions for a ring which is a subdirect sum of rings A_{i} having these properties to be faithfully represented on the right side of its left socle. We show ((4.F), p. 42) that up to technical trivia the rings A_{i} are matrix rings of the form
[...]. Each Q_{j} comes from the faithful irreducible matrix representation of a certain skewfield over a fixed skewfield D. The bottom row is filled in by arbitrary elements of D.
In Part V we construct an interesting class of rings faithfully represented on their left socle from a given partial ordering on a finite set, given skewfields, and given additive groups. This class of rings contains the ones in which every principal indecomposable left ideal has a unique minimal left ideal. We identify the uniquely determined subdirect summands mentioned above in terms of the given partial ordering (Proposition (5.2)). We conjecture that this technique serves to construct all the rings which are a unique subdirect sum of rings each having the property that every principalindecomposable left ideal contains a unique minimal left ideal.
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:  4 April 1966  
Funders: 
 
Record Number:  CaltechTHESIS:09252015104533327  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:09252015104533327  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9176  
Collection:  CaltechTHESIS  
Deposited By:  Benjamin Perez  
Deposited On:  25 Sep 2015 21:59  
Last Modified:  25 Sep 2015 21:59 
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