2025/08/02 by Sanjiv Subba, Subba, Sanjiv, Tikaram Subedi +1
Computer Science · Mathematics · #16S36 #16S50 #16U80 #16U99 #Advanced Algebra and Logic #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2508.01333
openalex publication_date 2025/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper investigates key properties of ZINC rings and their relationships with semicommutative and weakly semicommutative rings. We call an element x of a ring R zero insertive if x=arb for some a,b,r∈ R such that ab=0 and ZI(R) denotes the set of all zero insertive elements of R. We establish that a ring R is semicommutative if and only if ZI(R) ⊆ E(R), and weakly semicommutative if and only if ZI(R) ⊆ N(R), where E(R) and N(R) denote respectively the sets of idempotent elements and nilpotent elements. For ZINC rings with no nontrivial idempotents, ZI(R) ⊆ N(R). We prove that a finite direct product of ZINC rings is ZINC if and only if each component ring is ZINC, while an infinite direct product may fail to be ZINC. For n ≥ 2, if Mn(R) is ZINC, then R is weakly clean, however, the converse is not true (e.g., ℤ). Additionally, Mn(K) is ZINC for a division ring K if and only if K ≅ \mathbbF2. We, also, present a ZINC ring whose polynomial and power series extensions are not ZINC.