Rings in which every ideal disjoint with \(S\) is \(S\)-almost prime

Let \(R\) be a commutative ring with identity and \(S\) a multiplicative subset of \(R\). In this study, we introduce the concept of rings in which every ideal disjoint with \(S\) is \(S\)-almost prime. We investigate the possible transfer of the above ring property in the quotient rings, localizations, direct products, trivial ring extensions, and amalgamation algebra.