Generalized central idempotents in polynomial and power series rings

We give an explicit recursive description of idempotents in polynomial rings, formal power series rings, and Laurent polynomial rings over an arbitrary ring R. Using this description, we derive precise forms for idempotents that are central, left/right semicentral, inner/outer Peirce trivial, or Peirce trivial. As a consequence, we prove that R is abelian (resp. semiabelian, q-abelian) if and only if the corresponding extension ring R[x], R[[x]], or R[x,x−1] is abelian (resp. semiabelian, q-abelian). In particular, every Peirce trivial idempotent in R[x] or R[x,x−1] is conjugate to its constant term, holding true even without the assumption that R is a 2-primal ring. When R[x] (or R[x,x−1] ) is an ID-ring, we further show that any Peirce trivial idempotent matrix over such an extension is conjugate to a diagonal idempotent matrix over R. To support these structural outcomes, concrete matrix examples are introduced to validate the strict inclusions among these generalized idempotent classes and confirm that the implications in our framework are non-reversible.