International Journal of Applied Mathematics ————————————————————– Volume 25 No. 6 2012, 857-860 STRONGLY F-CLEAN RINGS Shervin Sahebi 1 , Mahya Derakhshan 2 § Department of Mathematics Islamic Azad University Central Tehran Branch, Tehran, IRAN e-mail 1 : sahebi@iauctb.ac.ir e-mail 2 : derakhshan.mahya@gmail.com Abstract: Let R be an associative ring with identity. R is said to be strongly f-clean ring if every element of R is the sum of an idempotent and a full element which commute. We study various properties of the strongly f-clean rings. AMS Subject Classification: 16D70, 16U99 Key Words: full elements, strongly clean rings, strongly f-clean rings 1. Introduction Let R be an associative ring with identity. An element in a ring is called clean, if it is the sum of an idempotent and a unit. An element a in a ring R is called strongly clean if a = e + u where e 2 = e ∈ R and u is a unit of R such that eu = ue. A ring R is called clean (resp. strongly clean) if every element of R is clean (resp. strongly clean). The f-clean rings were introduced by Li and Feng [4]. An element a ∈ R is said to be f-clean if it can be written as the sum of an idempotent and a full element. An element w ∈ R is said to be a full element if there exist s, t ∈ R such that swt = 1. A ring R is called f-clean if every element of R is f-clean. In this paper, we extend f-clean elements and introduce the concept of strongly f-clean elements. We study various properties of the strongly f-clean rings. Received: November 21, 2012 c  2012 Academic Publications § Correspondence author