Hurwitz series rings satisfying a zero divisor property

2023/12/17 by Behrooz Mosallaei, Moein Afrouzmehr, Mosallaei, Behrooz +4
Computer Science · Mathematics · #Advanced Algebra and Logic #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2312.10844

openalex publication_date 2023/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study zero divisors in Hurwitz series rings and Hurwitz polynomial rings over general noncommutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of Hurwitz series type (so reduced) ring R with a.c.c. on annihilator ideals, HR (the Hurwitz series ring with coefficients over R) has finitely many minimal prime ideals, say B₁, …, B_m such that B₁ ⋅ … ⋅ B_m = 0 and B_i = HA_i for some minimal prime ideal A_i of R for all i, where A₁, …, A_m are all minimal prime ideals of R. Additionally, we construct various types of (Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.

Related

• Prime and maximal ideals in Hurwitz polynomial rings
• Some combinatorial properties of the Hurwitz series ring
• A note on lower nil M-Armendariz rings
• On the Weierstrass Preparation Theorem over General Rings
• Prime ideals of Moh and the characteristic of the field