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BASIC CONCEPTS l
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abelian group algebraic extension Assume Bezout domain cancellation ideal class group closure completely integrally closed conditions are equivalent convex subgroup Corollary Dedekind domain defined denotes direct sum discrete valuation ring divisor domain with identity domain with quotient Exercise finite subset finitely generated ideal following conditions fractional ideal Hence homomorphism idempotent implies integral domain integral ideal integrally closed domain intersection invertible isomorphic Kronecker function ring Krull domain lattice-ordered Lemma maximal ideal monic polynomial Moreover Noetherian nonunit nonzero element nonzero ideal order isomorphic P-primary polynomial positive integer Priifer domain primary ideals principal ideal PROOF proper ideal proper prime ideal Proposition Prove Prufer quotient field rank one discrete regular element ring with identity semivaluation set of indeterminates set of maximal set of prime shows subgroup of G subring Theorem total quotient ring