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.