This research introduces novel center-like subsets, denoted as \(C^{\star }_{1} (\mathfrak {S}, \Gamma )\) , \(C_{2} (\mathfrak {S}, \Gamma )\) , and \(C^{\star }_{2} (\mathfrak {S}, \Gamma )\) , where \(\mathfrak {S}\) represents the ring and \(\Gamma \) denoted the (L,R)-generalized derivation. The study presents an innovative derivation for previously established center-like subsets and examines the interrelationships between these sets. Moreover, this investigation extends existing theorems by employing (L,R)-generalized derivations, in contrast to the conventional derivations utilized in prior research. The work also provides original proofs for a variety of center-like sets. To substantiate the necessity of the proposed hypotheses, the paper is supplemented with illustrative examples.
