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 \(a, b \in R\) $a,b\in R$ , \(a^m b \in I\) $a^{m}b\in I$ implies either \(a^n \in I\) $a^n\in I$ or \(b \in I\) $b\in I$ . 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.