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A generalization of Jónsson modules over commutative rings with identity
Oman, Gregory Grant

2006, Doctor of Philosophy, Ohio State University, Mathematics.
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 Jónsson algebra provided every subalgebra (B,F|B) has smaller cardinality than A (see [CK],[Co]). A Jónsson 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 Jónsson groups are the quasi-cyclic groups ℤ(p). Kurosh asked about the existence of a Jónsson group of size ℵ1, 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 Jónsson 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 Jónsson modules.

APA Citation

Oman, G. (2006). A generalization of Jónsson modules over commutative rings with identity. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

MLA Citation

Oman, Gregory. "A generalization of Jónsson modules over commutative rings with identity." Electronic Thesis or Dissertation. Ohio State University, 2006. OhioLINK Electronic Theses and Dissertations Center. 20 Jan 2018.

Chicago Citation

Oman, Gregory "A generalization of Jónsson modules over commutative rings with identity." Electronic Thesis or Dissertation. Ohio State University, 2006. https://etd.ohiolink.edu/