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This paper is a study of relations between selected classes of ideals in a domain D with unit. An ideal A in D is a valuation ideal if it is the intersection of D with an ideal of a valuation ring Rv containing D and contained in the quotient field of D. In this definition, if 1 'valuation' ' is replaced by Priifer or Dedekind, we obtain definitions for Priifer ideal or Dedekind Ideal, respectively. It is shown, in the first chapter, that every Priifer ideal in D is primary if and only if‘ there exists only one proper prime ideal in D, and that the set of Priifer ideals in D coincides with the set of primary ideals in D if and only, if D is a rank one valuation ring. The condition that prime ideals of D are chained is necessary and sufficient for Priifer ideals to be semi-primary; furthermore, the set of Priifer ideals coincides with the set of semi-primary ideals if and only if D is a valuation ring. It is also shown that Priifer ideals are powers of prime ideals if and only if the domain D is contained in only one valuation ring, it being P-adic for some prime ideal P of D. In the second chapter, it is shown that D is a Dedekind domain if every proper ideal of D is a Dedekind ideal, not necessarily for the same Dedekind domain. Several

Some containment relations between classes of ideals in an integral domain