Note on UN-rings

Let R be an associative ring with identity. Then R is said to be a UN-ring if every nonunit element of R can be written as a product of a unit and nilpotent elements. If in every UN-decomposition, the unit and nilpotent commute, the ring R is said to be a strongly UN-ring. In this paper, we study the notions of UN-rings in different contexts of commutative rings such us pullbacks, trivial ring extensions and amalgamations of algebras along ideals. Our aim is to generate new families of UN-rings. Namely, constructing UN-rings issued from pullbacks where the starting rings are not necessarily UN-ring and also to enrich the literature with such a rings. Examples illustrating the aims and scopes of our results are given.