On φ-(n,N )-ideals of Commutative Rings

Let R be a commutative ring with nonzero identity and n be a positive integer. In this paper, we introduce and investigate a new subclass of phi-n-absorbing primary ideals, which are called phi-(n,N)-ideals. Let phi: J(R) -> J (R) boolean OR {(sic)} be a function, where J (R) denotes the set of all ideals of R. A proper ideal I of R is called a phi-(n,N)-ideal if x(1) center dot center dot center dot x(n+1) is an element of I\phi(I) and x(1) center dot center dot center dot x(n) is not an element of I imply that the product of x(n+1) with (n - 1) of x(1) center dot center dot center dot, x(n) is in root 0 for all x(1), center dot center dot center dot ,x(n+1) is an element of R. In addition to giving many properties of phi-(n;N)-ideals, we also use the concept of phi-(n;N)-ideals to characterize rings that have only finitely many minimal prime ideals.