CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS

Authors : Ahmed Ayache, David E Dobbs

Pages : 99-114

Doi:10.24330/ieja.266230

View : 11 | Download : 11

Publication Date : 2014-12-01

Article Type : Research Paper

Abstract :Let R be a insert ignore into journalissuearticles values(commutative integral); domain with quotient K; let R0 be the integral closure of R insert ignore into journalissuearticles values(in K);. Then each overring of R insert ignore into journalissuearticles values(inside K); is a going-down domain if and only if R0 is a locally pseudo-valuation domain, T ⊆T0 satisfies going-down for every overring T of R, and tr. deg[VR0 insert ignore into journalissuearticles values(M);/Minsert ignore into journalissuearticles values(R0);M :R0/M] ≤ 1 for every maximal ideal M of R0 insert ignore into journalissuearticles values(where VR0 insert ignore into journalissuearticles values(M); denotes the valuation domain that is canonically associated to the pseudo-valuation domain insert ignore into journalissuearticles values(R0);M);. Additional equivalences are given in case R is locally finitedimensional. Applications include the case where R is integrally closed or R is not a Jaffard domain or R[X] is catenarian.
Keywords : Going down domain, overring, treed domain, pseudo valuation domain, valuation domain