2014/06/23 by Haifan Chen, Chen, H., Marjan Sheibani +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1406.5925
openalex publication_date 2014/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An element in a ring R is called uniquely weakly nil-clean if every element in R can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring in which every zero-divisor is uniquely weakly nil-clean is completely determined. We prove that every zero-divisor in a ring R is uniquely weakly nil-clean if and only if R is a D-ring, or R is abelian, periodic, and R/J(R) is isomorphic to a field F, \Bbb Z3⊕ \Bbb Z3, \Bbb Z3⊕ B where B is Boolean, or a Boolean ring. As a specific case, rings in which every zero-divisor a or -a is a nilpotent or an idempotent are also considered. Furthermore, we prove that every zero-divisor in a ring R is uniquely nil-clean if and only if R is a D-ring, or R is abelian, periodic; and R/J(R) is Boolean.\vskip3mm \no \bf Key words: Zero-divisor; Uniquely weakly nil-clean ring; Uniquely nil-clean ring.