A weak form of faithfulness, depending on Green’s equivalence $\mathcal{D}$, is introduced for a ring $R$ graded by a semigroup $S$. Suppose that $R$ satisfies this condition. It is shown that if $e$ and $f$ are $\mathcal{D}$-equivalent idempotents of $S$ and $R_e$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_f$ is semiprime (respectively, prime, semiprimitive, right primitive). In addition, it is shown that if $G$ and $H$ are maximal subgroups of $S$ lying in the same $\mathcal{D}$-class and $R_G$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_H$ is semiprime (respectively, prime, semiprimitive, right primitive).

AMS 2000 Mathematics subject classification: Primary 16W50. Secondary 20M25