On s-clean and s-nıl-clean rıngs

Let A be a commutative ring with identity. An element a G A is said to be S-clean (resp., S-nil-clean), where S ⊂ A is a given multiplicative set, if there exists s ∈ S such that sa is clean (resp.. nil-clean). The ring A is said to be S-clean (resp., S-nil-clcan) if each element of A is S-clean (resp., S-nil-clean). It is clear that every clean (resp., nil-clean) unital ring is S-clean (resp., S-nil-clean) and every homomorphic image of S-clean (resp., S-nil-clean) ring is S'-clean (resp., S'-nil-clean) ring where S' is the homomorphic image of S with 0 ∈ S'. In this manuscript, we investigate the transfer of this notion in amalgamation of rings introduced and studied by D'Anna, Finocchiaro and Fontana [8], in trivial ring extension and in pullback. Our attempt is to provide original and new classes of these rings.