Let D be a Dedekind domain with infinitely many maximal ideals, all of finite index, and K its quotient field. LetGiven any finite multiset {k 1 , . . . , kn} of integers greater than 1, we construct a polynomial in Int(D) which has exactly n essentially different factorizations into irreducibles in Int(D), the lengths of these factorizations being k 1 , . . . , kn. We also show that there is no transfer homomorphism from the multiplicative monoid of Int(D) to a block monoid.2010 Mathematics Subject Classification. 13A05; 13B25, 13F20, 11R04, 11C08.
An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ringinteger-valued polynomials on a principal ideal domain D with quotient field K, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator.2010 Mathematics Subject Classification. 13A05, 13B25, 13F20, 11R09, 11C08.
Let R be a commutative ring with identity. An element r ∈ R is said to be absolutely irreducible in R if for all natural numbers n > 1, r n has essentially only one factorization namely r n = r · · · r. If r ∈ R is irreducible in R but for some n > 1, r n has other factorizations distinct from r n = r · · · r, then r is called non-absolutely irreducible.In this paper, we construct non-absolutely irreducible elements in the ring Int(Z) = {f ∈ Q[x] | f (Z) ⊆ Z} of integer-valued polynomials. We also give generalizations of these constructions.2010 Mathematics Subject Classification. 13A05, 13B25, 13F20, 11R09, 11C08. Key words and phrases. irreducible elements, absolutely irreducible elements, non-absolutely irreducible elements, integer-valued polynomials.S. Nakato is supported by the Austrian Science Fund (FWF): P 30934.
In order to fully understand the factorization behavior of the ring Int(ℤ) = {f ∈ ℚ[x] | f (ℤ) ⊆ ℤ} of integer-valued polynomials on ℤ, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to recognize whether an integer-valued polynomial g/d] is irreducible in the case where d is a square-free integer and g ∈ ℤ[x] has fixed divisor d. For integer-valued polynomials with arbitrary composite denominators, so far there is no algorithmic criterion known to recognize whether they are irreducible. We describe a computational method which allows us to recognize all irreduciblexc polynomials in Int(ℤ). We present some known facts, preliminary new results and open questions.
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