Conf dr. Claudiu Wolf "Aritmetica in inele"

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CUPRINS

I. Aritmetică în inele integre.................................................................................................. 3 I.1 Divizibilitate................................................................................................................. 3 I.2 Inele euclidiene .......................................................................................................... 11 I.3 Inele de întregi pătratici, euclidiene faţă de normă .................................................... 14 I.4 Inele principale........................................................................................................... 17 I.5 Inele factoriale............................................................................................................ 20 I.6 Aritmetică în inele de polinoame ............................................................................... 25 Exerciţii ........................................................................................................................... 30 II. Module ............................................................................................................................ 36 II.1 Module, submodule, morfisme ................................................................................. 36 Exerciţii ........................................................................................................................... 46 II.2 Module factor şi teoreme de izomorfism .................................................................. 48 Exerciţii ........................................................................................................................... 52 II.3 Sume şi produse directe. Şiruri exacte...................................................................... 53 Exerciţii ........................................................................................................................... 66 II.4 Module libere ............................................................................................................ 68 Exerciţii ........................................................................................................................... 76 II.5 Bimodule, module duale ........................................................................................... 77 Exerciţii ........................................................................................................................... 81 III. Module finit generate peste inele principale ................................................................. 83 III.1 Submodulele unui modul liber de rang finit............................................................ 83 Exerciţii ........................................................................................................................... 89 III.2 Structura modulelor finit generate peste un inel principal ...................................... 91 Exerciţii ........................................................................................................................... 96 III.3 Module indecompozabile finit generate .................................................................. 97 Exerciţii ......................................................................................................................... 103 III.4 Aplicaţie: endomorfismele unui spaţiu vectorial finit dimensional....................... 104 1


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