Abstract
Let [Formula: see text] be a ring with identity. An element [Formula: see text] is called [Formula: see text]-clean if it can be written as the sum of an idempotent and a von Neumann regular element of [Formula: see text]; the ring [Formula: see text] is said to be [Formula: see text]-clean if every element of [Formula: see text] is [Formula: see text]-clean. Given a ring homomorphism [Formula: see text] and an ideal [Formula: see text], the amalgamation of [Formula: see text] and [Formula: see text] along [Formula: see text] with respect to [Formula: see text], denoted [Formula: see text], is a canonical subring of [Formula: see text]. In this paper, we investigate [Formula: see text]-clean elements and [Formula: see text]-clean rings in the setting of the amalgamation [Formula: see text]. We establish necessary and sufficient conditions for [Formula: see text] to be [Formula: see text]-clean and examine how the [Formula: see text]-clean property transfers between [Formula: see text], [Formula: see text], and the amalgamated ring. Furthermore, we study local and von Neumann local behavior, as well as [Formula: see text]-regularity and the exchange property of the amalgamation [Formula: see text].