If $x^n \in R$ for some $n \in \mathbb{N}$, do all of the homogeneous components $x_i$ of $x$ also satisfy $x_i^n \in R$?

Question

Let $R \subseteq S$ be an extension of $\mathbb{N}$-graded rings. Let $x = x_0 + x_1 + \dots + x_m$ ($x_i \in S_i$) satisfy $x^n \in R$ for some $n \in \mathbb{N}$. Does it follow that $x_i^n \in R$ for all $i$?

Certainly $x_0^n \in R$ and $x_m^n \in R$ because these are homogeneous components of $x^n \in R$. Does the same follow for the rest of the homogeneous components?

Answer 1

The answer is negative. If $S=\mathbb C[X_1,X_2,X_3,Y_1,Y_2,Y_3]$ we consider $x=X_1+X_2X_3+Y_1Y_2Y_3$ (a sum of three homogeneous elements of degree 1, 2, resp. 3) and let $$R=\mathbb C[X_1^2,X_1X_2X_3,X_2^2X_3^2+2X_1Y_1Y_2Y_3,X_2X_3Y_1Y_2Y_3,Y_1^2Y_2^2Y_3^2].$$ We have $x^2\in R$, but $X_2^2X_3^2\notin R$.