Let R be a commutative ring with unity \((1\not =0).\) We recall that a proper ideal I of R is called a strongly 1-absorbing primary ideal of R, if whenever \(abc\in I\) for some nonunit elements \(a, b, c\in R\) , then \(ab\in I\) or \(c\in \sqrt{0}\) . In this paper, we introduce a new class of ideals that is a generalization of the class of strongly 1-absorbing primary ideals. Let S be a multiplicative subset of R such that \(1\in S\) and \(0\not \in S\) . Let I be a proper ideal of R such that I disjoint with S, that is, \(S\cap I=\emptyset \) , for some \(s\in S\) , I is called a strongly S-1-absorbing primary ideal of R associated to s, if whenever \(abc\in I\) for some nonunit elements \(a, b, c\in R\) , then \(sab\in I\) or \(sc\in \sqrt{0}\) . This new concept is introduced as a subclass of the class of S-1-absorbing primary ideals and as a generalization to the class of strongly 1-absorbing primary ideals. In this paper, we have presented a range of different examples, properties, and characterizations of this new class of ideals. Moreover, we investigate basic properties of strongly S-1-absorbing primary ideals. Also, we use strongly S-1-absorbing primary ideals to characterize quasi-local rings with exactly one maximal ideal. Many other results are given to disclose the relations between this new concept and the S-primary ideals and the S-1-absorbing primary ideals. Finally, we introduce and study the strongly S-1-absorbing primary ideals of the quotient rings, the polynomial rings and rings of the form \(R(+)M.\)
