On the Integral Closure of Going-Down Rings

If R is an n–dimensional going-down ring (resp., an n–dimensional locally divided ring), with 0 < n < ∞, in which each zero-divisor is nilpotent and if P is a prime ideal of R of height n − 1 such that P ⊆ J(R), then the integral closure of R in R_P is a going-down ring (resp., a locally divided ring). Consequently, the question of whether the integral closure of a two-dimensional going-down domain R is a going-down domain is reduced to the subcase in which R is a divided domain that is integrally closed in R_P , where P is the unique height 1 prime ideal of R. The question of whether the integral closure of a going-down domain is a going-down domain is shown to be equivalent to the question of whether the integral closure of a going-down ring (resp., locally divided ring) in which each zero-divisor is nilpotent is a going-down ring (resp., locally divided ring).