Let R be an associative ring with unity. An element a in R is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von Neumann) element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. Further we prove that if 0 and 1 are the only idempotents in R, then an r-clean ring is an exchange ring. Also we show that the center of an r-clean ring is not necessary r-clean, but if 0 and 1 are the only idempotents in R, then the center of an r-clean ring is r-clean. Finally we give some properties and examples of r-clean rings
Ashrafi, N., Nasibi, E. (2013). Rings in which elements are the sum of an
idempotent and a regular element. Bulletin of the Iranian Mathematical Society, 39(3), 579-588.
MLA
N. Ashrafi; E. Nasibi. "Rings in which elements are the sum of an
idempotent and a regular element". Bulletin of the Iranian Mathematical Society, 39, 3, 2013, 579-588.
HARVARD
Ashrafi, N., Nasibi, E. (2013). 'Rings in which elements are the sum of an
idempotent and a regular element', Bulletin of the Iranian Mathematical Society, 39(3), pp. 579-588.
VANCOUVER
Ashrafi, N., Nasibi, E. Rings in which elements are the sum of an
idempotent and a regular element. Bulletin of the Iranian Mathematical Society, 2013; 39(3): 579-588.