FACTORIZATION IN SUBALGEBRAS OF THE POLYNOMIAL ALGEBRA

We investigate the factorization properties of subalgebras of the form K + fK[T] of the polynomial ring K[T], where f is in K[T], and the stability of these factorization properties under the standard operations. These subrings K + fK[T] satisfy the known generalizations of factoriality except possibly HFD and IDPF. Adomain is HFD if each non-zero element that is not a unit is a product of a unique number of irreducible elements. We prove that R = K + fK[T] is HFD if and only if R = K[T]. A domain is IDPF if forevery non-zero element a in the domain, the ascending sequence of sets of non-associate irreducible divisors of an stabilizes on a finite set. We prove that if the characteristic of K is zero, then R = K+ fK[T] is IDPF if and only if R = K[T]. When K has positive characteristic we prove that R is IDPF if and only if R = K[T] or f has only one root or K is algebraic over its prime subfield and f is a power of an irreducible polynomial. We compare the factorization properties of K + fK[T] with those of the subrings of the Gaussian integer.