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Modules which are isomorphic to their factor modules ASIYE SHAFIEYAN a shafieyan@math iut ac ir January 2015 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Atefeh Ghorbani a ghorbani@cc iut ac irAdvisor Dr Mahmood Behboodi mbehbood@cc iut ac ir2010 MSC Primary 13A99 13C05 Secondary 13E05 03E50 Keywords Almost Dedekind domain Artinian module Dedekind domain Discrete valuationring Noetherian module Strong limit cardinal Socle Uniserial module Abstract This M Sc thesis is based on the following paperGreg Oman Adam Salminen Modules Which are Isomorphic to Their Factor Modules Communications in Algebra 41 4 2013 1300 1315In this thesis all rings are commutative with identity element and all modules are unitary leftmodules unless indicated otherwise Let M be an R module Call M anti Hopfian providedM is not simple and M N M for every proper submodule N of M For example Pr fer group is anti Hopfian as a Z module First we show that every anti Hopfian module is unis erial Uniserial module denotes a module M whose submodules are linearly ordered by setinclusion Next we consider anti Hopfian modules over some special rings and we prove somegeneral results on anti Hopfian modules over commutative rings if R be a commutative ringand M be a nonsimple R module Then M is anti Hopfian if and only if the lattice of submod ules of M is isomorphic to 1 where is the first infinite ordinal Afterwards we turnour attention toward describing the anti Hopfian modules over discrete valuation rings almostDedekind domains and Dedekind domains Now let D be a Dedekind domain K the field offractions of D and P a prime ideal of D then we denote by C P the P component of K Dand it is defined to be the submodule of K D consisting of the elements of K D which arekilled by a power of P We show that every anti Hopfian module over a Dedekind domain isisomorphic to the module C P An infinite module M over a ring R is said to be homomorphically smaller HS for short overR if and only if M N M for every nonzero submodule N of M For example infinite fieldsand the ring Z of integers are HS as modules over themselves A modue M is HS over R if andonly if M m M for every nonzero m M All nonzero submodule of an HS module areHS Now we define HC modules Let R be a ring and let M be an infinite R module Call Mhomomorphically congruent HC for short provided M N M for every submodule N of M for which M N M Note that an HS R module is trivially HC also every anti HopfianR module is trivially HC In this article we study HC modules over commutative rings After afairly comprehensive review of the literature several natural examples are presented to motivateour 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 somegeneral results on HC modules Among other results we show that the annihilator of an HCmodule 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 theuniserial HC modules over a Noetherian ring Next we consider Noetherian and Artinian HCmodules Let M be an infinite faithful Artinian module over the domain D Suppose further

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