Weakly primal graded superideals

Let $G$ be an abelian group and let $R$ be a commutative $G$-graded super-ring (briefly, graded super-ring) with unity $1 ot=0$. We say that $ain h(R)$, where $h(R)$ is the set of homogeneous elements in $R$, is {it weakly prime} to a graded superideal $I$ of $R$ if $0 ot=rain I$, where $rin h(R)$, then $rin I$. If $ u(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 $P=igoplus_{gin G}( u(I)cap R_g^0+ u(I)cap R_g^1)cup{0}$ 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 $R$.