A NEW GENERALIZATION OF (m, n)-CLOSED IDEALS

Let R be a commutative ring with identity. For positive integers m and n , Anderson and Badawi ( Journal of Algebra and Its Applications 16(1):1750013 (21 pp), 2017) defined an ideal I of a ring R to be an ( m , n )-closed if whenever x m ∈ I , then x n ∈ I . In this paper we define and study a new generalization of the class of ( m , n )-closed ideals which is the class of quasi ( m , n )-closed ideals. A proper ideal I is called quasi ( m , n )-closed in R if for x ∈ R , x m ∈ I implies either x n ∈ I or x m - n ∈ I . That is, I is quasi ( m , n )-closed in R if and only if I is either ( m , n )-closed or ( m , m - n )-closed in R . We justify several properties and characterizations of quasi ( m , n )-closed ideals with many supporting examples. Furthermore, we investigate quasi ( m , n )-closed ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behavior of this class of ideals in idealization rings.