Let L(R) and K(R) be respectively the locally nilpotent radical and the upper nil radical of the ring R. If K(rho)G is a twisted group ring of a group G over a ring K and the order of every torsion element of G is not a zero divisor in K, then we prove that L(K)(rho)G subset of or equal to K(K(rho)G) subset of or equal to K(K)(rho)G. Moreover, let K(*)G be a crossed product of a group G over a prime ring K of characteristic p greater than or equal to 0. If the subgroup G(inn) has no p-elements when p > 0 and K(K) = 0, then K(K(*)G) = 0. [References: 7]
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