Strong rightD-domains |
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Authors: | Joachim Gräter |
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Affiliation: | (1) Institut für Algebra und Zahlentheorie, Technische Universität, Pockelsstrasse 14, D-3300 Braunschweig, Federal Republic of Germany |
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Abstract: | An integral domainR is called rightD-domain if its lattice of all right ideals is distributive. In § 2 a sufficient condition for an integral domainR is given such thatR is a rightD-domain if and only ifR is a leftD-domain. For example each integral domain which is algebraic over its center satisfies this criterion. Furthermore, a rightD-domain is called strong if its lattice of all fractional right ideals is distributive. Examples of strong rightD-domains are given in §4. Each overring of a strong rightD-domain is also a strong rightD-domain whereas arbitrary rightD-domains may have overrings which are no rightD-domains. Section 3 is mainly concerned with the set * of all left invertible fractional right ideals and the mapping :**,IIl–1 whereIl–1 denotes the left inverse ofI. For example, equivalent conditions are given for * to be a sublattice of and it is shown that is bijective if and only if (IJ)=(I)+(J) holds for allI,J*. Finally, §5 deals with (right)D-domains which are algebraic over their centersC. It is proved thatR is invariant if and only ifC is a commutative Prüfer domain andR the integral closure ofC inQ(R). |
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