Semigroup rings as weakly factorial domains, II

Let [Formula: see text] be an integral domain, [Formula: see text] be a nonzero torsionless commutative cancellative monoid with quotient group [Formula: see text], and [Formula: see text] be the semigroup ring of [Formula: see text] over [Formula: see text]. In this paper, among other things, we show that if [Formula: see text] (respectively, [Formula: see text], then [Formula: see text] is a weakly factorial domain if and only if [Formula: see text] is a weakly factorial GCD-domain, [Formula: see text] is a weakly factorial GCD-semigroup, and [Formula: see text] is of type [Formula: see text] (respectively, [Formula: see text] except [Formula: see text]).

Recent results on weakly factorial domains

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

Elasticity in Polynomial-Type Extensions

The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domain R and the elasticity of its polynomial ring R[x]. For example, if R has at least one atom, a sufficient condition for the polynomial ring R[x] to have elasticity 1 is that every non-constant irreducible polynomial f ∈ R[x] be irreducible in K[x]. We will determine the integral domains R whose polynomial rings satisfy this condition.