1. A class of uniquely (strongly) clean rings
- Author
-
Orhan Gürgün, Ayşe Çiğdem Özcan, and Matematik
- Subjects
Combinatorics ,Class (set theory) ,Ring (mathematics) ,Intersection ,Mathematics::Commutative Algebra ,General Mathematics ,Condensed Matter::Superconductivity ,Idempotence ,Element (category theory) ,Clean ring,strongly clean ring,uniquely clean ring,strongly J-clean ring ,Mathematics - Abstract
In this paper we call a ring R delta(r)-clean if every element is the sum of an idempotent and an element in delta(R-R) where delta(R-R) is the intersection of all essential maximal right ideals of R. If this representation is unique (and the elements commute) for every element we call the ring uniquely (strongly) delta(r)-clean. Various basic characterizations and properties of these rings are proved, and many extensions are investigated and many examples are given. In particular, we see that the class of delta(r)-clean rings lies between the class of uniquely clean rings and the class of exchange rings, and the class of uniquely strongly delta(r)-clean rings is a subclass of the class of uniquely strongly clean rings. We prove that R is delta(r)-clean if and only if R/delta(r)(R-R) is Boolean and R/Soc(R-R) is clean where Soc(R-R) is the right socle of R.
- Published
- 2014