Direct Product of a Graded Ring. Let $R$ be a graded ring, for example with nonnegative grading such that $$R=R_0 \oplus R_1 \oplus \dots$$ Is then $R^n= R \times \cdots \times R$ a graded ring (not with a trivial ...--prophetes.ai

Direct Product of a Graded Ring. Let $R$ be a graded ring, for example with nonnegative grading such that $$R=R_0 \oplus R_1 \oplus \dots$$ Is then $R^n= R \times \cdots \times R$ a graded ring (not with a trivial grading) ? If so, with what grading?

Sure. If $G$ is a monoid (in your case the natural numbers) and $R,S$ are $G$-graded rings, then $R \times S$ has the canonical $G$-grading $(R \times S)_g = \bigoplus_{a+b=g} R_a \times S_b$.