Bi-amalgamations subject to the nil-clean-like conditions

In this paper we introduce new classes of rings namely \(S_{1,n}\) $S_{1,n}$ nil-clean (resp., \(S_{n,2}\) $S_{n,2}$ nil-clean and \(S_{\sum ,2}\) $S_{\sum ,2}$ nil-clean) rings, and \(SI^n\) $S{I}^n$ rings as generalizations of nil-clean and boolean rings, respectively. We provide necessary and sufficient conditions for bi-amalgamations to inherit these properties. Our results are supported by illustrative examples arising as bi-amalgamations.