Unique factorization in rings with right ACC1

Unique Factorization in Generalized Power Series Rings

Let K be a field of characteristic zero and let K((R≤0)) denote the ring of generalized power series (i.e., formal sums with well-ordered support) with coefficients in K, and non-positive real exponents. Berarducci (2000) constructed an irreducible omnific integer, in the sense of Conway (2001), by first proving that an element of K((R≤0)) that is not divisible by a monomial and whose support h...

Unique Factorization in Regular Local Rings.

In this note we prove that every regular local ring of dimension 3 is a unique factorization domain. Nagata4 showed (Proposition 11) that if every regular local ring of dimension 3 is a unique factorization domain, then every regular local ring has unique factorization.* Thus, combining these results we have that every regular local ring is a unique factorization domain. Throughout this note R ...

Unique Factorization in Invariant Power Series Rings

Let G be a finite group, k a perfect field, and V a finite dimensional kG-module. We let G act on the power series k[[V ]] by linear substitutions and address the question of when the invariant power series k[[V ]] form a unique factorization domain. We prove that for a permutation module for a p-group in characteristic p, the answer is always positive. On the other hand, if G is a cyclic group...

Factorization in Polynomial Rings

We begin with some basic definitions. Definition 1.1. Let f, g ∈ F [x]. We say that f divides g, written f ∣∣g, if there exists an h ∈ F [x] such that g = fh, i.e. g is a multiple of f . Thus, for example, every f ∈ F [x] divides the zero polynomial 0, but g is divisible by 0 ⇐⇒ g = 0. By definition, f is a unit ⇐⇒ f ∣∣1. Recall also that the group of units (F [x])∗ of the ring F [x] is F ∗, th...

Factorization in Commutative Rings