Hurwitz series rings satisfying a zero divisor property

In this paper, we study zero divisors in the Hurwitz series rings and the Hurwitz polynomial rings over general non-commutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (the Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of the Hurwitz series type (so reduced) ring [Formula: see text] with [Formula: see text] on annihilator ideals, HR (the Hurwitz series ring with coefficients over [Formula: see text]) has finitely many minimal prime ideals, say [Formula: see text] such that [Formula: see text] and [Formula: see text] for some minimal prime ideal [Formula: see text] of [Formula: see text] for all [Formula: see text], where [Formula: see text] are all minimal prime ideals of [Formula: see text]. Additionally, we construct various types of (the Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.