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Rings faithfully represented on their left socle


Gordon, Robert (1966) Rings faithfully represented on their left socle. Dissertation (Ph.D.), California Institute of Technology.


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 K1,..., Kr such that the i,jth block of a typical matrix is a Ki x Kj matrix with arbitrary entries in a subgroup Dij of the additive group of a fixed skewfield D. Each Dii is a sub-skewfield of D and Dri = 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 Dij ≠ 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 Dij is a one-dimensional left vector space over Dii (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 A1,...,As of the same type. Namely, each Ai has only one isomorphism class of minimal left ideals and the minimal left ideals of different Ai are non-isomorphic as left A-modules. We give (Theorem (2.1)) necessary and sufficient conditions for a ring which is a subdirect sum of rings Ai 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 Ai are matrix rings of the form

[...]. Each Qj 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 principal-indecomposable 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):
  • Dade, E. C.
Thesis Committee:
  • Unknown, Unknown
Defense Date:4 April 1966
Funding AgencyGrant Number
National Defense Education ActUNSPECIFIED
Record Number:CaltechTHESIS:09252015-104533327
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9176
Deposited By: Benjamin Perez
Deposited On:25 Sep 2015 21:59
Last Modified:25 Sep 2015 21:59

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