Weakly primal graded superideals

Abstract

Let $G$ be an abelian group and let $R$ be a commutative $G$-graded super-ring (briefly, graded super-ring) with unity $1\not=0$. We say that $a\in h(R)$, where $h(R)$ is the set of homogeneous elements in $R$, is {\it weakly prime} to a graded superideal $I$ of $R$ if $0\not=ra\in I$, where $r\in h(R)$, then $r\in I$. If $\nu(I)$ is the set of homogeneous elements in $R$ that are not weakly prime to $I$, then we define $I$ to be weakly primal if $P=\bigoplus_{g\in G}(\nu(I)\cap R_g^0+\nu(I)\cap R_g^1)\cup\{0\}$ forms a graded superideal of $R$. In this paper we study weakly primal graded superideals of $R$. Moreover, we classify the relationship among the families of weakly prime graded superideals, primal and weakly primal graded superideals of $R$.