Strong right D-domains
An integral domain R (i.e. an associative ring without zerodivisors having a unit-element 1) is called right D-domain if its lattice of right ideals is distributive and commutative right D-domains are exactly the commutative Prüfer domains. In {\S}2 a sufficient condition for an integral domain R is given such that R is a right D-domain if and only if R is a left D-domain. For example each integral domain which is algebraic over its center satisfies this criterion. Furthermore, a right D-domain is called strong if its lattice of all fractional right ideals \({\mathcal R}\) is distributive. Examples of strong right D-domains are given in {\S}4. Each overring of a strong right D-domain is also a strong right D-domain whereas arbitrary right D-domains may have overrings which are not right D-domains. {\S}3 is mainly concerned with the set \({}^*{\mathcal R}\) of all left invertible fractional right ideals and the mapping \(\lambda\) : \({}^*{\mathcal R}\to {\mathcal L}^*\), \(I\mapsto I_ l^{-1}\) where \(I_ l^{-1}\) denotes the left inverse of I. For example, equivalent conditions are given for \({}^*{\mathcal R}\) to be a sublattice of \({\mathcal R}\) and it is shown that \(\lambda\) is bijective if and only if \(\lambda (I\cap J)=\lambda (I)+\lambda (J)\) holds for all \(I,J\in^*{\mathcal R}\). Finally, {\S}5 deals with (right) D-domains which are algebraic over their centers C. It is proved that C is invariant if and only if C is a commutative Prüfer domain and R the integral closure of C in Q(R).
scientific article; zbMATH DE number 4112800