A generalization of primary ideals of commutative rings

Let [Formula: see text] be a commutative ring and [Formula: see text] an endomorphism of [Formula: see text]. In this paper we introduce and study the concept of [Formula: see text]-primary ideal which is a generalization of primary ideal. An ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-primary if whenever [Formula: see text] for some [Formula: see text] then [Formula: see text] or [Formula: see text]. We show that [Formula: see text]-primary ideals share many analogous properties of primary ideals and we also describe the behavior of [Formula: see text]-primary property across various ring theoretic constructions such as trivial ring extensions and amalgamations.