Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of 1-absorbing primary ideals. Let m, n be nonzero positive integers such that \(m > n\) . A proper ideal I of R is said to be an (m, n)-absorbing primary ideal if whenever nonunit elements \(a_1,...,a_{m} \in R\) and \(a_1...a_{m}\in I\) , then \(a_1...a_{n} \in I\) or \(a_{n+1}...a_m\in \sqrt{I}\) . Some properties of (m, n)-absorbing primary ideals are investigated. For example, we show that if R admits an (m, n)-absorbing primary ideal that is not an \((m-1,n-1)\) -absorbing primary ideal, then R is a quasilocal ring. We give an example of a (4, 3)-absorbing primary ideal of a ring R, that is not a 1-absorbing primary ideal of R. We have studied this concept in the chained, divided and Dedekind ring. We give some basic properties of this class of ideals and we study (m, n)-absorbing primary ideals of localization of rings, direct product of rings, trivial ring extension and amalgamation of rings. A proper ideal I of R is called an U-(m, n)-absorbing primary ideal of R if whenever \(a_1...a_{m}\in I\) , for some elements \(a_1,...,a_{m} \in R\) , then \(a_1...a_{n} \in I\) or \(a_{n+1}...a_m\in \sqrt{I}\) , we study some connections between (m, n)-absorbing primary ideal and U-(m, n)-absorbing primary ideal.
