Author
Oman, Gregory

Year
2006

Advisor
Loper, Alan

Abstract
In universal algebra, an algebra is defined to be a pair (A,F) consisting of a set A and a collection F of finitary operations on A. If F is countable, then the algebra (A,F) is called a Jonsson algebra provided every subalgebra (B,F|B) has smaller cardinality than A (see [CK],[Co]). A Jonsson group then is a group G in which all proper subgroups have smaller cardinality than G. W.R. Scott showed in [Sc1] that the only abelian Jonsson groups are the quasi-cyclic groups. Kurosh asked about the existence of a Jonsson group of size aleph 1, and Shelah gave an affirmative answer in [Sh]. This piqued the interest of commutative algebraists Gilmer and Heinzer. They call an infinite module M a Jonsson module if every proper submodule of M has smaller cardinality than M. We investigate the consequences of a weaker notion. Instead of requiring every submodule of M of the same cardinality as M to be equal to M, we only require that it be isomorphic to M. We call such modules congruent. In this dissertation, we develop the theory of congruent modules over commutative rings with identity and use this theory to give some new results on Jonsson modules.

Thesis
Oman, Gregory Grant.pdf