In this thesis , all rings are commutative with identity element and all modules are unitary left modules unless indicated otherwise . Let M be an R-module . Call M anti-Hopfian provided M is not simple and M ?M/N for every proper submodule N of M . For example Prüfer group is anti-Hopfian as a -module . if R be a commutative ring and M be a nonsimple R-module Then M is anti-Hopfian if and only if the lattice of submodules of M is isomorphic to ?+1 (where ? is the first infinite ordinal) . Afterwards we turn our attention toward describing the anti-Hopfian modules over discrete valuation rings , almost Dedekind domains , and Dedekind domains . We show that every anti-Hopfian module over a Dedekind domain is isomorphic to the module C( ). An infinite module M over a ring R is said to be homomorphically smaller (HS for short) over R if and only if |M/N| |M| for every nonzero submodule N of M. For example infinite fields and the ring of integers are HS as modules over themselves . A modue M is HS over R if and only if |M/(m)| |M| for every nonzero m ? M . All nonzero submodule of an HS module are HS . Now we define HC modules . Let R be a ring and let M be an infinite R-module . Call M homomorphically congruent (HC for short) provided M ?M/N for every submodule N of M for which |M/N|=|M| . Note that an HS R-module is trivially HC , also every anti-Hopfian R-module is trivially HC . In this article , we study HC modules over commutative rings . After a fairly comprehensive review of the literature , several natural examples are presented to motivate our study . We begin with a first example which characterizes the HC vector spaces over a field . Let F be a field , and V be an infinite F-vector space . Then V is HC if and only if dim(V)=1 or |V| |F|. In the second example we characterize the HC abelian groups . We then prove some general results on HC modules . Among other results , we show that the annihilator of an HC module is a prime ideal . also we prove that every HC module is either torsion or torsion-free . Next we show that the torsion-free HC modules are precisely the HS modules . Afterwards , we turn our attention toward describing the uniserial HC modules . We then characterize the uniserial HC modules over a Noetherian ring . Next we consider Noetherian and Artinian HC modules . Let M be an infinite faithful Artinian module over the domain D . Suppose further that the socle of M is simple . Then M is HC if and only if M is anti-Hopfian . But we do not know if we need the assumption that the socle of M is simple to deduce that M is anti-Hopfian . We finish this section by using our results to give HC module-theoretic characterizations of fields . We also provide a characterization of the HC modules over a Dedekind domain . We first classify the torsion HC modules over an arbitrary Dedekind domain (which is not a field) . We next complete our description of the HC modules over the Dedekind domains and consider the torsion-free HC modules over a Dedekind domain . Finally , we close with some open questions .
