| In a 1932 paper, Dorroh developed a technique to embed an algebra over a field in an algebra with identity. In 1946, Brown and McCoy commented that Dorroh's extension technique could be used for arbitrary rings, not just algebras over fields. They went on to show that a slight generalization of the Dorroh extension provides a means by which ally ring can be embedded in a ring with identity with the same characteristic. This dissertation addresses the following question: if a ring has ring property P , can it be embedded in a ring with identity which has property P ?;Generalized Dorroh extensions are first examined and their ideal structure described. A particular homomorphic image of a generalized Dorroh extension, the Szendrei extension, is also introduced, and it is shown that the Szendrei extension can be used to preserve some ring properties that cannot be preserved through any generalized Dorroh extension. Other extension techniques are briefly mentioned, including Everett extensions which are a generalization of both Dorroh and Szendrei extensions.;Next, Amitsur-Kurosh radicals are discussed along with how radicals of a ring relate to radicals of extensions of that ring, particularly generalized Dorroh extensions. One main result is that for any Amitsur-Kurosh radical gamma and any ring E which contains (all isomorphic copy of) a ring R as an ideal, gamma(E) = gamma( R) if and only if gamma(E/R) = 0. A description of the nilradicals of a generalized Dorroh extension is also given.;After that, the question of whether or not specific ring properties can be passed on to sonic extension with identity is addressed. It is shown that although many ring properties can be preserved, there exist examples of rings with a specific property which cannot be embedded in any ring with identity which has that property. Finally, rings with additional structure are examined, e.g., rings with involution and normed rings. It is shown that both Dorroh and Szendrei extensions frequently possess complementary additional structure. |
