Lipman, Joseph On complete ideals in regular local rings. (English) Zbl 0693.13011 Algebraic geometry and commutative algebra, Vol. I, 203-231 (1988). [For the entire collection see Zbl 0655.00011.] In appendix 5 of volume II of a famous text he coauthored, O. Zariski proved that in a 2-dimensional regular local ring, every complete (i.e., integrally closed) ideal can be uniquely factored into a product of simple complete ideals. He also states that his arguments do not seem to generalize to higher dimension, at least not without substantial revision. The present author, using a different approach has found a partial generalization of Zariski’s result. He shows that in a regular local ring R of arbitrary degree \(at\quad least\quad 2,\) there is an interesting set of complete ideals (those with finite support) for which a unique factorization of sorts does hold. When applied to the 2-dimensional case, Zariski’s result is recovered. Although space prohibits more detail here, this paper is fairly easy to read, and is well worth the effort. Reviewer: S.McAdam Cited in 11 ReviewsCited in 15 Documents MSC: 13H05 Regular local rings 13A15 Ideals and multiplicative ideal theory in commutative rings Keywords:integrally closed ideals; regular local ring; complete ideals Citations:Zbl 0655.00011 PDF BibTeX XML OpenURL