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 < infinity, in which each zero-divisor is nilpotent and if P is a prime ideal of R of height n - 1 such that P subset of or equal to 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).