Strong rightD-domains

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 ℛ 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 *ℛ of all left invertible fractional right ideals and the mapping λ:*ℛ→ℒ*,I↦Il−1 whereIl−1 denotes the left inverse ofI. For example, equivalent conditions are given for *ℛ to be a sublattice of ℛ and it is shown that λ is bijective if and only if λ(I▸J)=λ(I)+λ(J) holds for allI,J∈*ℛ. 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 Prufer domain andR the integral closure ofC inQ(R).