Is Kernel of a map from free module to finitely generated graded module graded?

Question

Let $R$ be a Noetherian (or $K$-algebra for some field $K$.) Assume that $R$ is graded by some abelian group. Let $M$ be a finitely generated $R$-module. Then, we may assume $M=Rm_{1} \oplus \cdots \oplus Rm_{n}$ for some $m_{i} \in M$, hence we have a kernel of the surjective map $R^{n} \to M$. My question is that, is the kernel graded so that we can write it as $R/I_{1} \oplus \cdots \oplus R/I_{n}$? I know this holds for PID, however, I don't know whether it is true or not for UFD or any higher cases. It would be appreciated to give any example/counterexamples and the condition the above statement should be true.