On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Abstract

Abstract. 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.

Small and Large Ideals of an Associative Ring

“…As evidence, the residually finite rings have not been classified. Moreover, for every uncountable cardinal κ, there exist non-Noetherian valuation rings V of cardinality κ for which every nonzero left ideal of V is large (see Theorem 2.8 of [19]).…”

“…Say that M is homomorphically smaller (HS for short) if and only if |M/N | < |M | for all nonzero submodules N of M . Various structural results on HS modules were obtained in [19]. Dually, infinite modules M (over a commutative ring) for which |N | < |M | for all proper submodules N of M have also received attention in the literature.…”

Residually small commutative rings

“…Conversely, assume that R is not a field. By Proposition 3.2 of [17], Ann(R) := {r ∈ R : rR = {0}} is a prime ideal of R. Since R has an identity, we see that Ann(R) = {0}. Therefore, R is a domain.…”

“…Definition 1 is also closely related to the concept of a homomorphically smaller module. In Oman & Salminen [17] the authors, borrowing terminology introduced in Tucci [22], define a (unitary) module M over a commutative ring R to be homomorphically smaller (HS for short) if and only if M is infinite and |M/N | < |M | for every nonzero submodule N of M . We also define a ring R to be an HS ring if R is an HS module over itself, that is, if R is infinite and |R/I| < |R| for every nonzero ideal I of R. The definition of an HS ring extends the notion of (infinite, commutative) residually finite rings appearing in Chew & Lawn [4] 1 .…”

Jónsson and HS Modules over Commutative Rings

“…Following Tucci's terminology, Salminen and the author define an infinite module over a ring to be -smaller (HS for short) provided | / | < | | for every nonzero submodule of . Some structural results on these modules were obtained in Oman and Salminen [36] and Oman [37]. We exposit the theory of HS modules in Sections 7-9.…”