Prime radicals of graded $\Omega$-groups

In this paper, we introduce the class of graded $\Omega$-groups, which includes: groups; associative, conformal and vertex algebras; Lie algebras and graded algebras. The graded prime radical of a graded $\Omega$-group is defined, and its elementwise characterization is given. It is shown that the graded prime radical of a graded $\Omega$-groups with a finiteness condition coincides with the lower weakly solvable (in Parfyonov sense) radical.