Add to ORKGAbstract. Let R be an associative ring with unity. An element a ∈ 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. Furthermore, 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. 1
AbstractA ring R is called clean if every element is the sum of an idempotent and a unit, and R is c...
The conditions that allow an element of an associative, unital, not necessarily commutative ring R, ...
A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a unit and a pro...
AbstractAn associative ring with unity is called clean (respectively uniquely clean) if every elemen...
A ring is said to be clean if each element in the ring can be written as the sum of a unit and an id...
AbstractAn element in a ring R is said to be clean (respectively unit-regular) if it is the sum (res...
Let R be an associative unital ring and let a R be a strongly nil clean element. We introduce a new ...
summary:Let $R$ be an associative unital ring and let $a\in R$ be a strongly nil clean element. We i...
summary:An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an id...
AbstractA ring R with identity is called strongly clean if every element of R is the sum of an idemp...
AbstractA ring is called uniquely clean if every element is uniquely the sum of an idempotent and a ...
Let R be an associative ring with identity. Let Id(R) and U(R) denote the set of idempotents and th...
A ring R is called n-clean ring if every element of R can be written as a sum of an idempotent and n...
A ring R is almost unit-clean provided that every element in R is equivalent to the sum of an idempo...
A clean ring is one in which every element is a sum of an idempotent and a unit. It is shown that ev...
AbstractA ring R is called clean if every element is the sum of an idempotent and a unit, and R is c...
The conditions that allow an element of an associative, unital, not necessarily commutative ring R, ...
A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a unit and a pro...
AbstractAn associative ring with unity is called clean (respectively uniquely clean) if every elemen...
A ring is said to be clean if each element in the ring can be written as the sum of a unit and an id...
AbstractAn element in a ring R is said to be clean (respectively unit-regular) if it is the sum (res...
Let R be an associative unital ring and let a R be a strongly nil clean element. We introduce a new ...
summary:Let $R$ be an associative unital ring and let $a\in R$ be a strongly nil clean element. We i...
summary:An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an id...
AbstractA ring R with identity is called strongly clean if every element of R is the sum of an idemp...
AbstractA ring is called uniquely clean if every element is uniquely the sum of an idempotent and a ...
Let R be an associative ring with identity. Let Id(R) and U(R) denote the set of idempotents and th...
A ring R is called n-clean ring if every element of R can be written as a sum of an idempotent and n...
A ring R is almost unit-clean provided that every element in R is equivalent to the sum of an idempo...
A clean ring is one in which every element is a sum of an idempotent and a unit. It is shown that ev...
AbstractA ring R is called clean if every element is the sum of an idempotent and a unit, and R is c...
The conditions that allow an element of an associative, unital, not necessarily commutative ring R, ...
A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a unit and a pro...