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 Local radical and semisimple classes of rings
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Local radical and semisimple classes of rings Stewart, Patrick Noble
Abstract
For any cardinal number K ≥2 and any nonempty class of rings ℛ we make the following definitions. The class ℛ(K) is the class of all rings R such that every subring of R which is generated by a set of cardinality strictly less than K is in ℛ . The class ℛg(K) is the class of all rings R such that every nonzero homomorphic image of R contains a nonzero subring in ℛ which is generated by a set of cardinality strictly less than K . Several properties of the classes ℛg(K) and ℛ(K) are determined. In particular, conditions are specified which imply that ℛ(K) is a radical class or a semisimple class. Necessary and sufficient conditions that the class ℑ of all ℛg(K) semisimple rings be equal to ℑ(K) are given. The classes ℛ(K) and ℛg(K) when K = 2 or K = (formula omitted)₀ are considered in detail for various classesℛ (including the cases when ℛ is one of the wellknown radical classes). In all cases when ℛ is one of the wellknown radical classes it Is shown that ℛ(2) and ℛ(formula omitted) are radical classes and whenever they contain all nilpotent rings they are shown to be special radical classes. Those radical classes ℛ(2) which are contained in FC (R € FC if and only if for all x € R , x is torsion) are characterized. Let ℛ be any radical class. The largest radical class (formula omitted) (if one exists) such that (formula omitted)(R) Ո ℛ(R) = (0) for all rings R is defined to be the local complement of ℛ̅̅ and is denoted by ℛ. If ℛ = ℛ(formula omitted) then the local complement ℛ exists and ℛ= ℛ(2) . The local complements of all radicals discussed are determined. We are able to apply some of these results in order to classify those classes of rings which are both semisimple and radical classes.
Item Metadata
Title  Local radical and semisimple classes of rings 
Creator  Stewart, Patrick Noble 
Publisher  University of British Columbia 
Date Issued  1969 
Description 
For any cardinal number K ≥2 and any nonempty class of rings ℛ we make the following definitions. The class ℛ(K) is the class of all rings R such that every subring of R which is generated by a set of cardinality strictly less than K is in ℛ . The class ℛg(K) is the class of all rings R such that every nonzero homomorphic image of R contains a nonzero subring in ℛ which is generated by a set of cardinality strictly less than K .
Several properties of the classes ℛg(K) and ℛ(K) are determined. In particular, conditions are specified which imply that ℛ(K) is a radical class or a semisimple class. Necessary and sufficient conditions that the class ℑ of all ℛg(K) semisimple rings be equal to ℑ(K) are given.
The classes ℛ(K) and ℛg(K) when K = 2 or K = (formula omitted)₀ are considered in detail for various classesℛ (including the cases when ℛ is one of the wellknown radical classes). In all cases when ℛ is one of the wellknown radical classes it Is shown that ℛ(2) and ℛ(formula omitted) are radical classes and whenever they contain all nilpotent rings they are shown to be special radical classes. Those radical classes ℛ(2) which are contained in FC (R € FC if and only if for all x € R , x is torsion) are characterized. Let ℛ be any radical class. The largest radical class (formula omitted) (if one exists) such that (formula omitted)(R) Ո ℛ(R) = (0) for all rings R is defined to be the local complement of ℛ̅̅ and is denoted by ℛ. If ℛ = ℛ(formula omitted) then the local complement ℛ exists and ℛ= ℛ(2) . The local complements of all radicals discussed are determined.
We are able to apply some of these results in order to classify those classes of rings which are both semisimple and radical classes.

Subject  Rings (Algebra) 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110621 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0302247 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.