On graded pseudo 2-prime ideals (Q6586855)

Let \(R\) be a commutative ring with a nonzero identity, \(M\) a unital \(R\)-module, and \(G\) a commutative additive monoid with an identity element 0. We say that \(R\) is a graded ring (or \(G\)-graded ring) if \(R = \bigoplus_{g \in G} R_g\), where the \(R_g\) are additive subgroups of \(R\) and satisfy \(R_g R_h \subseteq R_{g+h}\) for all \(g, h \in G\). The elements of \(R_g\) are termed homogeneous elements of degree \(g\), and the collection of all homogeneous elements of \(R\) is denoted by \(h(R)\). An ideal \(I\) of \(R\) is considered a homogeneous ideal if \(I = \bigoplus_{g \in G} (I \cap R_g)\). A proper graded ideal \(I\) of \(R\) is referred to as a graded prime ideal if whenever \(ab \in I\) for some \(a, b \in h(R)\), then either \(a \in I\) or \(b \in I\). A proper graded ideal \(I\) is called a graded 2-absorbing ideal if \(abc \in I\) for some \(a, b, c \in h(R)\) implies \(ab \in I\), \(ac \in I\), or \(bc \in I\). A proper graded ideal \(I\) is designated as a graded 2-prime ideal if \(ab \in I\) for some \(a, b \in h(R)\) leads to \(a^2 \in I\) or \(b^2 \in I\). Additionally, a proper graded ideal \(I\) is termed a graded pseudo 2-prime ideal if \(ab \in I\) for some \(a, b \in h(R)\) implies there exists \(n \in \mathbb{N}\) such that \(a^{2n} \in I^n\) or \(b^{2n} \in I^n\). This paper discusses the relationships between graded pseudo 2-prime ideals and various classical graded ideals, including graded 2-prime ideals, graded 2-absorbing ideals, and graded irreducible ideals. The authors also examined the stability of graded pseudo 2-prime ideals under graded homomorphisms, in factor rings, in the localization of graded rings, in Cartesian products of graded rings, and in graded trivial extensions. Finally, the authors analyzed finite unions of graded pseudo 2-prime ideals and proved the graded pseudo 2-prime avoidance theorem.