On strongly quasi S-primary ideals

This paper centers around one of several generalizations of primary ideals. Which is an intermediate class between S-primary ideals and quasi S-primary ideals. Let R be a commutative ring with identity and S be a multiplicative closed subset of R. A proper ideal I of R disjoint from S is called strongly quasi S-primary if there exists an s is an element of S such that whenever x, y is an element of R and xy is an element of I, then either sx(2) is an element of I or sy is an element of root I. Many basic properties of strongly quasi S-primary ideals are given, and examples are presented to distinguish the last concept from other classical ideals. Moreover, forms of strongly quasi S-primary ideals in polynomial rings, power series rings and idealization of a module are investigated.