Generalization of primal superideals

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 $\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\varnothing\}$ be any function, where $\mathfrak{I}(R)$ denotes the set of all proper superideals of $R$. A homogeneous element $a\in R$ is $\phi$-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 $\phi$-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 $\phi$-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.