Properties of \(\phi\)-primal graded ideals (Q2421745)

Summary: Let \(R\) be a commutative graded ring with unity \(1 \ne 0\). A proper graded ideal of \(R\) is a graded ideal \(I\) of \(R\) such that \(I \neq R\). Let \(\phi : \mathfrak{I}(R) \rightarrow \mathfrak{I}(R) \cup\{\emptyset\}\) be any function, where \(\mathfrak{I}(R)\) denotes the set of all proper graded ideals of \(R\). A homogeneous element \(a \in R\) is \(\phi\)-\textit{prime} to \(I\) if \(r a \in I - \phi(I)\) where \(r\) is a homogeneous element in \(R\); then \(r \in I\). An element \(a = \sum_{g \in G} a_g \in R\) is \(\phi\)-prime to \(I\) if at least one component \(a_g\) of \(a\) is \(\phi\)-prime to \(I\). Therefore, \(a = \sum_{g \in G} a_g \in R\) is not \(\phi\)-prime to \(I\) if each component \(a_g\) of \(a\) is not \(\phi\)-prime to \(I\). We denote by \(\nu_\phi(I)\) the set of all elements in \(R\) that are not \(\phi\)-prime to \(I\). We define \(I\) to be \(\phi\)-\textit{primal} if the set \(P = \nu_\phi(I) + \phi(I)\) (if \(\phi \neq \phi_\emptyset\)) or \(P = \nu_\phi(I)\) (if \(\phi = \phi_\emptyset\)) forms a graded ideal of \(R\). In the work by the author [Algebra Discrete Math. 21, No. 2, 202--213 (2016; Zbl 1357.13003)], the author studied the generalization of primal superideals over a commutative super-ring \(R\) with unity. In this paper we generalize the work by \textit{A. Jaber} [Algebra Discrete Math. 21, No. 2, 202--213 (2016; Zbl 1357.13003)] to the graded case and we study more properties about this generalization.