2017/02/22 by Ajay K. Sharma, Dhiren Kumar Basnet, Sharma, Ajay +1
Mathematics · #Rings, Modules, and Algebras #Commutative Algebra and Its Applications #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.1702.06734
Motivated by the concept of weakly clean rings, we introduce the concept of weakly r-clean rings. We define an element x of a ring R as weakly r-clean if it can be expressed as x=r+e or x=r-e where e is an idempotent and r is a regular element of R. If all the elements of R are weakly r-clean then R is called a weakly r-clean ring. We discuss some of its properties in this article. Also we generalise this concept of weakly r-clean ring to weakly g(x)-r-clean ring, where g(x) ∈ C(R)[x] and C(R) is the centre of the ring R. Finally we introduce the concept of weakly ⋆-clean and ⋆-r-clean ring and discuss some of their properties.