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A ring theoretic approach to radicals of extensionsWilliams, Jessica Lynn 01 May 2015 (has links)
The Jacobson radical of a ring was first formally studied in 1945 by Nathan Jacobson and is an important object in modern abstract algebra. The analogous notion of the Jacobson radical for a module is referred to as the radical of a module. The radical of a module is the intersection of all its maximal submodules. In general, the radical of a module is simpler than the module itself and contains important information about the module. The study of the radical of a module often appears as an incidental to other investigations.
This thesis represents work towards understanding the radical of a module extension. Given a ring $R$ and $R$modules $A,B,X$ such that $X$ is an extension of $B$ by $A$ as in the short exact sequence $$0 rightarrow A rightarrow X rightarrow B rightarrow 0 ,$$ we seek to determine properties of the radical of $X$, denoted $rad{X}$. These properties are dependent on the ring $R$ and properties of the modules $A$ and $B$.
In this thesis we examine several different types of extensions and discuss a phenomenon in which a nonzero radical implies a split sequence. We work in the context of rings and their ideals. Extensions of abelian groups provide motivation for the results we prove about injectivity of radicals of extensions involving divisible modules and torsion modules. We are able to prove such properties of the radical for extensions of modules over principal ideal domains and Dedekind domains. Expanding upon these cases, we explore a more general construction of an extension and use it to explain our motivating abelian group results. We use the theorems proven about this construction to remark on possible generalizations to other types of rings and modules. We conclude with plans to generalize our statements by translating into terms of infinite matrices and $h$local rings.

12 
Gas phase cyclisation and rearrangement reactions of aromatic free radicalsHutchison, Helen Susan January 1987 (has links)
No description available.

13 
Studies on Lewis acidpromoted radical cyclization reactions顧深, Gu, Shen. January 2002 (has links)
published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

14 
Approaches towards the development of novel, clean radical deoxygenations and cyclisationsJones, Leighton January 2003 (has links)
No description available.

15 
Kinetic and spectroscopic studies of radicalcations in aqueous solutionWarren, C. J. January 1987 (has links)
No description available.

16 
A radical cyclisation approach to the synthesis of NheterocyclesPettifer, Robert M. January 1998 (has links)
No description available.

17 
The coppercatalyzed oxidation of biologically relevant thiolsSilvester, Stephen January 1999 (has links)
No description available.

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Thermochemistry of cyclic and acyclic radicals and kinetics of their recombination reactionsGuiver, S. C. January 1986 (has links)
No description available.

19 
Stereoselective Hex5enyl radical cyclisationsSmith, Elin Hopcyn January 1994 (has links)
No description available.

20 
The synthesis and polymerization of methacrylate macronomersTopping, Clare January 1998 (has links)
No description available.

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