Jónsson Modules over Noetherian Rings

Abstract

Let R be a commutative ring with identity, and let M be an infinite unitary R-module. M is said to be a Jónsson module provided every proper submodule of M has strictly smaller cardinality than M. Utilizing earlier results of the author [11 Oman , G. ( 2009 ). Some results on Jónsson modules over a commutative ring . Houston J. Math. 35 ( 1 ): 1 – 12 .[Web of Science ®] , [Google Scholar]] as well as results of Gilmer/Heinzer, Weakley, and Heinzer/Lantz [8 Gilmer , R. , Heinzer , W. ( 1983 ). On Jónsson modules over a commutative ring . Acta Sci. Math. 46 : 3 – 15 . [Google Scholar], 10 Heinzer , W. , Lantz , D. ( 1985 ). Artinian modules and modules of which all proper submodules are finitely generated . J. Algebra 95 : 201 – 216 .[Crossref], [Web of Science ®] , [Google Scholar], 14 Weakley , W. ( 1983 ). Almost finitely generated modules . J. Algebra 84 : 189 – 219 .[Crossref], [Web of Science ®] , [Google Scholar]], we study Jónsson modules over Noetherian rings. After a brief introduction, we classify the countable Jónsson modules over an arbitrary ring up to quotient equivalence. We then give a complete description of the Jónsson modules over a 1-dimensional Noetherian ring, extending W. R. Scott's classification over ℤ. We show that these results may be extended to Jónsson modules over an arbitrary Noetherian ring if one assumes The Generalized Continuum Hypothesis. Finally, we close with a list of open problems.