The Cleanness of the Subrings of M₂ (Zₚ)

Let be a ring. Ring is said to be a clean ring if every element of R can be expressed as the sum of a unit and an idempotent element. Furthermore, there are r-clean rings. An r-clean ring is a generalization of a clean ring. In an r-clean ring, all of its elements can be represented as the sum of a regular element and an idempotent element. Moreover, strongly r-clean rings were introduced. A strongly r-clean ring is a ring where every element of the ring can be expressed as the sum of a regular and an idempotent element, and the multiplication of that regular and idempotent is commutative. On the other hand, there is a ring of the set of matrices over ring denotes by . In this paper, we will discuss the cleanness properties, especially strongly r-clean of the subring of . The aims of this paper are to find the characteristics of strongly r-clean of the subring of . Here, we assumed that is a ring of matrix over .