What question did this study set out to answer?

The central aim is to investigate the S-clean and S-nil-clean properties of rings, particularly their transfer in certain ring constructions.

February 2, 2026

On s-clean and s-nıl-clean rıngs

Key Points

The central aim is to investigate the S-clean and S-nil-clean properties of rings, particularly their transfer in certain ring constructions.
Defined S-clean and S-nil-clean elements and rings in terms of a multiplicative set S.
Analyzed the implications of these definitions in amalgamations, trivial ring extensions, and pullbacks.
Explored new classes of S-clean and S-nil-clean rings based on the existing literature.
Established that every clean and nil-clean unital ring is also S-clean and S-nil-clean.
Showed that homomorphic images preserve S-clean and S-nil-clean properties under certain conditions.
Proposed original classes of S-clean and S-nil-clean rings that expand current understanding.

Abstract

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. © 2024 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.

Mark Helpful

Bookmark

Relay