Regularity and R-Decomposition

Let (Formula presented.) be a complex unital Banach algebra and (Formula presented.) be a regularity. In order to characterize the accumulation point of the spectrum of (Formula presented.) associated with (Formula presented.), we define (Formula presented.) -decomposition of elements in (Formula presented.), and give the conditions to make (Formula presented.) a regularity, where (Formula presented.) is the set of elements with (Formula presented.) -decomposition. Further, we study the relation between the spectrum of (Formula presented.) associated with (Formula presented.) and the accumulation point of the spectrum of (Formula presented.) associated with (Formula presented.). Besides, some properties of (Formula presented.) are discussed.