JONSSON MODULES OVER NOETHERIAN RINGS

Let R be a commutative ring with identity, and let M be an infinite unitary R-module. M is said to be a Jonsson module provided every proper submodule of M has strictly smaller cardinality than M. Utilizing earlier results of the author [11] as well as results of GilmerlHeinzer, Weakley, and HeinzerlLantz [8, 10, 14], we study Jonsson modules over Noetherian rings. After a brief introduction, we classify the countable Jonsson modules over an arbitrary ring up to quotient equivalence. We then give a complete description of the Jonsson modules over a 1-dimensional Noetherian ring, extending W. R. Scott's classification over Z. We show that these results may be extended to Jonsson modules over an arbitrary Noetherian ring if one assumes The Generalized Continuum Hypothesis. Finally, we close with a list of open problems.