On graded-(m, n)-prime ideals of commutative graded rings

Let G be an abelian group written addditively and R be a commutative graded ring of type G with identity and m, n be positive integers. The main purpose of this paper is to introduce the class of graded-(m, n)-prime ideals which lies properly between the classes of graded-prime and graded-(m, n)-closed ideals introduced recently by the authors in Ahmed et al.(Moroccan J Algebra Geom Appl 1(2):1-10, 2022). A proper graded ideal I of R is called graded-(m, n)-prime if for some homogeneous elements a, b is an element of R, a(m)b is an element of I implies either an is an element of Iorb is an element of I. Several characterizations of this new class of graded ideals with several original examples are given. Moreover, we defend the actions of graded-(m, n)-prime ideals in several extensions of graded rings, especially in idealization of graded modules and amalgamation of graded rings and similarly to graded-primary decomposition, we introduce the graded-(m, n)-decomposition of graded ideals and we prove that every graded ideal in a graded-n-Noetherian ring has a graded-(m, n)-decomposition. Finally, the graded-(m, n)-prime avoidance theorem is given.