Communications of the
Korean Mathematical Society CKMS

An element in a ring $R$ with identity is called invo-clean if it is the sum of an idempotent and an involution and $R$ is called invo-clean if every element of $R$ is invo-clean. Let $C(R)$ be the center of a ring $R$ and $g(x)$ be a fixed polynomial in $C(R)[x]$. We introduce the new notion of $g(x)$-invo clean. $R$ is called $g(x)$-invo if every element in $R$ is a sum of an involution and a root of $g(x)$. In this paper, we investigate many properties and examples of $g(x)$-invo clean rings. Moreover, we characterize invo-clean as $g(x)$-invo clean rings where $g(x)=(x-a)(x-b)$, $a,b\in C(R)$ and $b-a\in Inv(R)$. Finally, some classes of $g(x)$-invo clean rings are discussed.

Keywords: Invo-clean ring, $g(x)$-invo clean ring, unitly invo-clean ring, amalgamated algebra

MSC numbers: 13A99, 16D60, 16U99