Generalization of primal superideals

Abstract

Let \(R\) be a commutative super-ring with unity \(1\not=0\). A proper superideal of \(R\) is a superideal \(I\) of \(R\) such that \(I\not=R\). Let \(\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\emptyset\}\) be any function, where \(\mathfrak{I}(R)\) denotes the set of all proper superideals of \(R\). A homogeneous element \(a\in R\) is \(\phi\)-{\it prime} to \(I\) if \(ra\in I-\phi(I)\) where \(r\) is a homogeneous element in \(R\), then \(r\in I\). We denote by \(\nu_\phi(I)\) the set of all homogeneous elements in \(R\) that are not \(\phi\)-prime to \(I\). We define \(I\) to be \(\phi\)-\textit{primal} if the set \[P=\begin{cases}[(\nu_\phi(I))_0+(\nu_\phi(I))_1\cup\{0\}]+\phi(I) & :\quad {\rm if}\ \phi\not=\phi_\emptyset\\ (\nu_\phi(I))_0+(\nu_\phi(I))_1& :\quad {\rm if}\ \phi=\phi_\emptyset\end{cases}\]
forms a superideal of \(R\). For example if we take \(\phi_\emptyset(I)=\emptyset\) (resp. \(\phi_0(I)=0\)), a \(\phi\)-\textit{primal} superideal is a primal superideal (resp., a weakly primal superideal). In this paper we study several generalizations of primal superideals of \(R\) and their properties.