Resumen de Some results on Jónsson modules over a commutative ring

Greg Oman

• Let M be an infinite unitary module over a commutative ring R with identity. M is called Jonsson over R provided every proper submodule of M has smaller cardinality than M; M is large if M has cardinality larger than R. Extending results of Gilmer and Heinzer, we prove that if M is Jonsson over R, then either M is isomorphic to R and R is a field, or M is a torsion module. We show that there are no large Jonsson modules of regular or singular strong limit cardinality. In particular, the Generalized Continuum Hypothesis (GCH) implies there are no large Jonsson modules. Necessary and sufficient conditions are given for an infinitely generated Jonsson module to be countable. As applications, we prove there are no large uniserial or Artinian modules. Under the GCH, we derive a new characterization of the quasi-cyclic groups.