Half-factorial subrings of factorial domains

A half-factorial domain D is a domain in which every non-zero element that is not a unit is a product of a unique number of irreducible elements of D. We characterize half-factorial subrings R of factorial domains S when S is the integral closure of R and their unit groups are identical. Let A   be a factorial domain and A[T]A[T] the polynomial ring over A in the variable T. The characterization is used to describe the half-factorial A-subalgebras R   with multiplicative conductors of A[T]A[T] into R.