Journal of the
Korean Mathematical Society JKMS

An associative ring $R$ with identity is called a clean ring if every element of $R$ is the sum of a unit and an idempotent. In this paper, we introduce the concept of $f$-clean rings. We study various properties of $f$-clean rings. Let $C=\left(\begin{smallmatrix} A & V \\ W & B \\ \end{smallmatrix} \right)$ be a Morita Context ring. We determine conditions under which the ring $C$ is $f$-clean. Moreover, we introduce the concept of rings having many full elements. We investigate characterizations of this kind of rings and show that rings having many full elements are closed under matrix rings and Morita Context rings.

Keywords: full elements, $f$-clean rings, matrix rings, rings having any full elements

MSC numbers: Primary 16D70; 16U99