Elementary Equivalence of Categories of Modules over Rings, Endomorphism Rings, and Automorphism Groups of Modules

1 B asic N otions from the Set T heory,M odelT heory,and C ategory T heory 6 1.1 FirstO rderLanguages . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. 6 1.2 Axiom sand BasicNotionsoftheTheory NBG . .. .. .. . .. .. .. . .. .. .. . .. .. .. 9 1.3 M odels,Deducibility,and Elem entary Equivalence . .. .. . .. .. .. . .. .. .. . .. .. .. 12 1.4 Ultra lters,Ultraproducts,and Ultrapowers .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. 15 1.5 BasicNotionsfrom theCategory Theory.Category ofM odulesoverRings .. .. .. . .. .. .. 16