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osu1164331653.pdf (409.69 KB)
ETD Abstract Container
Abstract Header
A generalization of Jónsson modules over commutative rings with identity
Author Info
Oman, Gregory Grant
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1164331653
Abstract Details
Year and Degree
2006, Doctor of Philosophy, Ohio State University, Mathematics.
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 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 ℵ
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 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.
Committee
Kenneth Loper (Advisor)
Subject Headings
Mathematics
Recommended Citations
Refworks
EndNote
RIS
Mendeley
Citations
Oman, G. G. (2006).
A generalization of Jónsson modules over commutative rings with identity
[Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1164331653
APA Style (7th edition)
Oman, Gregory.
A generalization of Jónsson modules over commutative rings with identity.
2006. Ohio State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1164331653.
MLA Style (8th edition)
Oman, Gregory. "A generalization of Jónsson modules over commutative rings with identity." Doctoral dissertation, Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=osu1164331653
Chicago Manual of Style (17th edition)
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Document number:
osu1164331653
Download Count:
912
Copyright Info
© 2006, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.