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International Journal of Mathematics and Mathematical Sciences
Volume 2014 (2014), Article ID 120907, 13 pages
Research Article

Jónsson and HS Modules over Commutative Rings

Department of Mathematics, The University of Colorado, Colorado Springs, CO 80918, USA

Received 16 October 2013; Revised 14 January 2014; Accepted 20 January 2014; Published 12 March 2014

Academic Editor: A. Cigdem Ozcan

Copyright © 2014 Greg Oman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let be a commutative ring with identity and let be an infinite unitary -module. (Unless indicated otherwise, all rings are commutative with identity 10 and all modules are unitary.) Then is called a Jónsson module provided every proper submodule of has smaller cardinality than . Dually, is said to be homomorphically smaller (HS for short) if for every nonzero submodule of . In this survey paper, we bring the reader up to speed on current research on these structures by presenting the principal results on Jónsson and HS modules. We conclude the paper with several open problems.