On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Abstract

AbstractLetRbe a commutative ring with identity, and letMbe a unitary module overR. We callMH-smaller (HS for short) if and only ifMis infinite and |M/N| < |M| for every nonzero submoduleNofM. 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: supposeMis faithful overR,Ris a domain (we will show that we can restrict to this case without loss of generality), andKis the quotient field ofR. IfMis HS overR, thenRis HS as a module over itself,R⊆M⊆K, and there exists a generating setSforMoverRwith |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.