Jónsson extensions of one-dimensional semilocal domains
A J-extension T of a commutative ring R is a commutative ring containing R, not finitely generated as an R algebra, but minimal with respect to this property, so that every R subalgebra of T unequal to T is finitely generated. The authors characterize the J-extensions T of a one- dimensional semilocal Noetherian domain R inside the field of fractions K of R: if \((.)^-\) denotes ``total-quotient-ring-of-completion'' then the minimal nonzero nilpotent ideals I of \(R^-\) correspond to such J- extensions T bijectively, with I the kernel of \(R^-\to T^-\) and T the intersection of K in R -/I with the completion R* of R modulo R*\(\cap I\). - This theorem also yields as a corollary the multiplicity mult(T) as mult(R)-mult(R*/P) (for P some minimal prime of R*). The authors further prove that if R is Gorenstein with local integral closure S, then there is at most one J-extension of R in K, and if also \(mult(R)=2\), then the rings between R and S are linarly ordered. Examples are given to show the hypotheses necessary.
scientific article; zbMATH DE number 4067057
scientific article