How can one describe the associated graded ring of $\mathbb Z_p$ with respect to its maximal ideal $p\mathbb Z_p$? If one follows the definition, the associated graded ring is $$\mathrm{gr}_{p\mathbb Z_p}\mathbb Z_p = \bigoplus_{n=0}^\infty p^n\mathbb Z_p/p^{n+1}\mathbb Z_p$$ Here each summand $p^n\mathbb Z_p/p^{n+1}\mathbb Z_p$ is isomorphic to the finite field $\mathbb F_p$. So additively the graded ring is just infinitely many copies of $\mathbb F_p$, and the multiplicative structure is defined with respect to the grading.
Is there some way to give a more conceptual description of the graded ring here? Is it perhaps isomorphic to some nice ring?