Let M be a commutative cancellative atomic monoid and M * its set of nonunits. Let ρ(x) denote the elasticity of factorization ofWe call an atomic integral domain D fully elastic if its multiplicative monoid, denoted D • , is fully elastic. We examine the full elasticity property in the context of Krull monoids with finite divisor class groups, numerical monoids and certain integral domains. For every real number α ≥ 1, we construct a fully elastic Dedekind domain D with ρ(D) = α. In particular, while we show that noncyclic numerical monoids are never fully elastic, we do verify that several large classes of Krull monoids, and hence certain Krull domains, are fully elastic.

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A connection is developed between polynomials invariant under abelian permutation of their variables and minimal zero sequences in a finite abelian group. This connection is exploited to count the number of minimal invariant polynomials for various abelian groups.