ROOT CLOSURE IN COMMUTATIVE RINGS OF THE FORM A[[X]]

Let A=(A(n))(n >= 0) be an ascending chain of commutative rings with identity, and let A[[X]] be the ring of power series with coefficient of degree i in A(i) for each i is an element of N. Thus, A[[X]] = {f = Sigma(n >= 0). a(n)X(n) is an element of A[[X]]/a(n) is an element of A(n) for all n is an element of N}. In this article, we consider a ring extension A[[X]] subset of B[[X]], where A=(A(n))(n >= 0) and B=(B-n)(n0) are two chains of commutative rings such that for each i is an element of N, there is a ring extension A(i) subset of B-i. We give necessary and sufficient conditions for A[[X]] to be seminormal, root closed, or t-closed in B[[X]].