In Kemper, A Course in Commutative Algebra there is an alternative definition of associated graded ring, as shown below.
Let $R$ be a Noetherian local ring with maximal ideal $m$. $$R^{*}:=R[m\cdot t]\subset R[t]$$ $$R_{d}^{*}:=R\cdot t^{d} \cap R^{*}=m^{d}\cdot t^{d}$$ $$\mathrm{gr}(R):=R^{*}/(m)_{R^{*}}$$ $$\mathrm{gr}(R)_{d}:=R_{d}^{*}/(R_{d}^{*}\cdot m)\cong m^{d}/m^{d+1}$$ then $\mathrm{gr}(R)=\bigoplus_{d\in \mathbb{N}} \mathrm{gr}(R)_{d}$.
But I don't understand why $\mathrm{gr}(R)$ defined in this way can be decomposed as direct sum like this. I don't see a clear inclusion here and I think it's not clear how to define multiplication in this way (like in the definition of graded ring, $S_{i}\cdot S_{j}\subset S_{i+j}$). Could you please show me in detail how $\mathrm{gr}(R)$ is decomposed into direct sum of $\mathrm{gr}(R)_{d}$ and the isomorphism in the fourth line?