ON GENERALIZATIONS OF PRIME IDEALS

Let R be a commutative ring with identity. Let phi : S(R) -> S(R) boolean OR {empty set} be a function, where S(R) is the set of ideals of R. Suppose n >= 2 is a positive integer. A nonzero proper ideal I of R is called (n - 1, n) - phi-prime if, whenever a(1), a(2), ..., a(n) is an element of R and a(1)a(2) ... a(n) is an element of I\phi(I), the product of (n - 1) of the a(i)'s is in I. In this article, we study (n - 1, n) - phi-prime ideals (n >= 2). A number of results concerning (n - 1, n) - phi-prime ideals and examples of (n - 1, n) - phi-prime ideals are also given. Finally, rings with the property that for some phi, every proper ideal is (n - 1, n) - phi-prime, are characterized.