An element $a$ in a ring $R$ is very clean in case there exists‎ ‎an idempotent $ein R$ such that $ae = ea$ and either $a‎- ‎e$ or $a‎ + ‎e$ is invertible‎. ‎An element $a$ in a ring $R$ is very $J$-clean‎ ‎provided that there exists an idempotent $ein R$ such that $ae =‎ ‎ea$ and either $a-ein J(R)$ or $a‎ + ‎ein J(R)$‎. ‎Let $R$ be a‎ ‎local ring‎, ‎and let $sin C(R)$‎. ‎We prove that $Ain K_...

A ring R is said to be n-clean if every element can be written as a sum of an idempotent and n units. The class of these rings contains clean ring and n-good rings in which each element is a sum of n units. In this paper, we show that for any ring R, the endomorphism ring of a free R-module of rank at least 2 is 2-clean and that the ring B(R) of all ω × ω row and column-finite matrices over any...

An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently, Anderson and Camillo (Anderson, D. D., Camillo, V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as we...

in this paper we focus on a special class of commutative local‎ ‎rings called spap-rings and study the relationship between this‎ ‎class and other classes of rings‎. ‎we characterize the structure of‎ ‎modules and especially‎, ‎the prime submodules of free modules over‎ ‎an spap-ring and derive some basic properties‎. ‎then we answer the‎ ‎question of lam and reyes about strongly oka ideals fam...

 In this paper we characterize all $2times 2$ idempotent and nilpotent matrices over an integral domain and then we characterize all $2times 2$ strongly nil-clean matrices over a PID. Also, we determine when a $2times 2$ matrix  over a UFD is nil-clean.

A ring R is called clean if every element of the sum a unit and an idempotent. Motivated by question proposed Lam on cleanness von Neumann Algebras, Vaš introduced more natural concept for ?-rings, ?-cleanness. More precisely, ?-ring ?- projection (?-invariant idempotent). Let F be finite field G abelian group. In this paper, we introduce two classes involutions group rings form characterize ?-...

In this paper our goal to thoroughly determine the rings in which each non-unit element is a product of nilpotent and quasi-idempotent.