For any pointed space , the mod Lannes-Zarati homomorphism is considered as a graded associated version of the mod Hurewicz map in the -term of the Adams spectral sequence. In this paper, we investigate the behavior of and for an odd prime .

We characterize the fixed divisor of a polynomial [Formula: see text] in [Formula: see text] by looking at the contraction of the powers of the maximal ideals of the overring [Formula: see text] containing [Formula: see text]. Given a prime and a positive integer , we also obtain a complete description of the ideal of polynomials in [Formula: see text] whose fixed divisor is divisible by [Formula: see text] in terms of its primary components.

Let [Formula: see text] be a holomorphy ring in a global field , and a classical maximal [Formula: see text]-order in a central simple algebra over . We study sets of lengths of factorizations of cancellative elements of into atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of [Formula: see text], which implies that all the structural finiteness results for sets of lengths-valid for commutative Krull monoids with finite class group-hold also true for . If [Formula: see text] is the ring of algebraic integers of a number field , we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.