Citation
Kruse, Robert Leroy (1964) Rings with periodic additive groups in which all subrings are ideals. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd09302002085204
Abstract
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A ring in which every subring is a two sided ideal is called a vring. This dissertation is a classification of all vrings with periodic additive group. It is first shown that a ring is a vring with periodic additive group if and only if it is the restricted ring direct sum of vrings whose additive groups are pgroups for different primes p. Such rings are called pvrings. It is next shown that a pvring must be nil, or be isomorphic to the ring of rational integers mod p[superscript n] for some n > 1, or be isomorphic to the direct sum of the prime field of p elements and a nil pvring.
The classification of nil pvrings constitutes the major part of this dissertation. Nil pvrings containing elements of unbounded additive order are first characterised. Redei has shown that for any element x of a nil pvring either (I) x[superscript 2] is a natural multiple of x or (II) px[superscript 2] is a natural multiple of x although x[superscript 2] is not a natural multiple of x. Because of this result it is possible to study a nil pvring possessing a bound on the additive orders of its elements by decomposing the ring into an additive group direct sum of cyclic groups. It is shown that aside from elements in the annihilator of the ring, there is a decomposition of the ring with at most two generators of type (I) and three of type (II). The possible defining relations for these nil pvrings are enumerated.
Item Type:  Thesis (Dissertation (Ph.D.)) 

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:  1 January 1964 
Record Number:  CaltechETD:etd09302002085204 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd09302002085204 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3824 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  30 Sep 2002 
Last Modified:  26 Dec 2012 03:03 
Thesis Files

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