## Theses and Dissertations

Dissertation

Spring 2015

#### Degree Name

PhD (Doctor of Philosophy)

Mathematics

Victor Camillo

Miodrag Iovanov

#### Abstract

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 non-zero 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.

#### Public Abstract

Abstract algebra is founded in the rigorous and axiomatic study of number systems used every day. One such number system is the integers. The set of all integers together with the operations of multiplication and addition are an example of an abstract algebraic object called a ring. Much of modern research in algebra is devoted to studying rings. A fundamental component of this research is the study of modules, which are abstract algebraic structures related to rings.

An important object is the Jacobson radical of a ring, first formally studied in 1945 by Nathan Jacobson. The radical of a module is the module theoretic analog to the Jacobson radical of a ring. Loosely, the radical of an object is the intersection of its maximal sub-objects. In general, the radical is simpler than the object itself and contains important information about the object.

This thesis begins with consideration of a specific and surprising example. The example is an extension of the rational numbers by a semisimple abelian group, which are both modules over the ring of integers. This extension has the property of splitting if and only if the radical is non-zero. This sparked further investigation, and this thesis presents results about radicals of similar extensions over rings with properties similar to those of the integers. This work is grounded in basic theory and therefore has potential to be applied in multiple areas of mathematical research.

vii, 77 pages

#### Bibliography

Includes bibliographical references (page 77).