Geodesic
Generalization of primal superideals
Algebra and discrete mathematics, Tome 21 (2016) no. 2, pp. 202-213.

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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.
Keywords: primal superideal, $\phi$-$P$-primal superideal, $\phi$-prime superideal. 
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     author = {Ameer Jaber},
     title = {Generalization of primal superideals},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2016_21_2_a3/}
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Ameer Jaber. Generalization of primal superideals. Algebra and discrete mathematics, Tome 21 (2016) no. 2, pp. 202-213. http://geodesic.mathdoc.fr/item/ADM_2016_21_2_a3/

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