On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

摘要

Let R be a commutative ring with identity, and let M be a unitary module over R. We call M H-smaller (HS for short) if and only if M is infinite and |M/N| < M for every nonzero submodule N of M. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose M is faithful over R, R is a domain (we will show that we can restrict to this case without loss of generality), and K is the quotient field of R. If M is HS over R, then R is HS as a module over itself, R ? M ? K, and there exists a generating set S for M over R with S < R. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.