مدول هاي تزريقي ضعيف روي دامنه هاي صحيح

چكيده انگليسي :

Weak injective Modules over Integral Domain Nafiseh Nazari n nazari@math iut ac ir January 18 2012 Master of Science Thesis Department of Mathematical Scienoes Isfahan University of Technology Isfahan 84156 83111 IranSupervisors Dr Mahmood Behboodi mbehboodi@cc iut ac ir 2000 MSC 13C11 Key word H divisible modules Pure injective modules Weak injective modules Matlis andEnochs cotorsion modules Weak injective global dimension Almost perfect domain Flat cover Weak injective envelope Weak dimension Abstract Weak injective modules over commutative rings have been defined by Lee as modules M whichsatisfy A M 0 for all R modules A of weak dimension 1 In this thesis based on 10 and 11 it is shown that class of modules weak dimension 1 and the class of weak injectivemodules form a cotorsion pair In general the class of weak injectives lies strictlybetween the classes of h divisible and injective R modules An R module D is called h divisible if itis an epic image of an injective R module H divisible R modules are always divisible but notinjective in general and Dedekind domains are characterized as those domains over which h divisibleR modules are injective It is proved that weak injective R modules coincide to injective R modules ifand only if R is a Pr fer domain Several features of weak injective modules are discussed in chapter4 Also it is established the existence of a universal test module for weak injectivity i e an R moduleU such that U N 0 for any R module N implies that N is weak injective We study weak injective envelopes of modules and we show that all R modules admit weak injective envelope i e every R module can be embedded in a weak injective module with cokernel of weak dimension 1 In particular it is shown that if R is a commutative domain and Q B R it is field of quotients then forthe weak injective envelope of an R module N it is a direct sum of copies of Q if and only if N is aflat R module Also it is shown that there is a close relation between the flat cover whose existenceis guaranteed by well known theorem of Bican El Bashir and Enochs 26 and the weak injectiveenvelope of any R module This yield a method of constructing weak injective envelopes from flatcovers and vice versa This relation can be best illustrated by the diagram in Theorem 5 35 Similarrelation exists between the Enochs cotorsion envelopes and the weak dimension 1 covers ofmodules Over a coherent domain the cokernel of an arbitrary Enochs cotorsion module in it is weak injective envelope is always pure injective A ring R almost perfect if all proper factor rings of R areperfect Almost perfect domains can be characterized in several ways one of which is that theconcepts of Matlis cotorsion and Enochs cotorsion coincide These domains have already been thetopics of several research papers One of our main purposes is to add new characterizations in termsof weak injectivity So it is introduced the global weak injective dimension of a domain R It turns outthat this dimension is equal to the supremum of projective dimension of R modules of weakdimension 1 A main result is Theorem 6 14 stating that a domain R is almost perfect if and only if itsglobal weak injective dimension is 1 This yields several other possibilities for characterizing almostperfect domains cf Corollary 6 20 H divisible pure injective R modules are always weak injective but the converse is not true whenever R is an almost perfect non Dedekind domain

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مدول هاي تزريقي ضعيف روي دامنه هاي صحيح

مدول h- بخش پذير , مدول هم-تاب ايناكس و هم-تاب متليس , بعد تزريقي ضعيف فراگير , دامنه تقريبا كامل , پوشش تخت