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By the general theory of invariants of finite groups, there exist $(n-1)!$ monomials homogeneous polynomials $u_1, \dots, u_p$ ($p=(n-1)!$) such that every element $f$ of $R^{Z_n}$ can be uniquely written $f = u_1 g_1 + \cdots+ u_p g_p$, where $g_1,\dots,g_p$ are symmetric functions. I don't know whether an explicit description of $u_1,\dots,u_p$ is known for arbitrary $n$.
By the general theory of invariants of finite groups, there exist $(n-1)!$ monomials $u_1, \dots, u_p$ ($p=(n-1)!$) such that every element $f$ of $R^{Z_n}$ can be uniquely written $f = u_1 g_1 + \cdots+ u_p g_p$, where $g_1,\dots,g_p$ are symmetric functions. I don't know whether an explicit description of $u_1,\dots,u_p$ is known for arbitrary $n$.