On m-clean index of ring

A ring [Formula: see text] is called [Formula: see text]-clean if each element of [Formula: see text] can be expressed as the sum of a unit and an [Formula: see text]-potent in [Formula: see text]. We introduce the concept of the [Formula: see text]-clean index associated with a ring [Formula: see text]. For any element [Formula: see text], we define [Formula: see text], where [Formula: see text] denotes the set of units of [Formula: see text]. Then the [Formula: see text]-clean index of the ring [Formula: see text] is defined as [Formula: see text] and it is denoted by [Formula: see text]-[Formula: see text]-[Formula: see text]. In this paper, we study some properties of this index and characterize the ring of [Formula: see text]-clean index [Formula: see text]. We show that the rings with the [Formula: see text]-clean index 1 are precisely the rings whose [Formula: see text]-clean index is [Formula: see text] and each [Formula: see text]-potent is idempotent. We establish some connections between [Formula: see text]-clean index of the ring [Formula: see text] and the underlying rings [Formula: see text] and [Formula: see text]-module [Formula: see text]. We also determine the [Formula: see text]-clean index of the trivial extension [Formula: see text] and the [Formula: see text]-clean index of the ideal extension [Formula: see text] with some associated conditions. Finally, we discuss the amalgamated structure of ring in connection to the [Formula: see text]-clean index.