Let $R$ be a commutative graded ring (https://en.wikipedia.org/wiki/Graded_ring) with unity ,
graded by a monoid (in particular $\mathbb Z$ or $\mathbb N \cup \{0\}$ if helpful for the purpose ) , let $M$ be a graded module over $R$ such that $M$ is finitely generated as an $R$-module , then does there exist a finite set of homogenous elements which also generates $M$ over $R$ ? If not true in general then what condition on $R$ or $M$ will ensure it ? Has there been any work on this ?
$M$ is a direct sum of its graded pieces (by definition), $M = \oplus M_i$, so if we denote $\pi_i : M \to M_i$ the $ith$ projection, then (by the defining properties of the direct sum):
1) for any $f \in M$, only finitely many $\pi_i(f)$ are nonzero.
2) for any $f \in M$, then $f = \Sigma \pi_i(f)$.
So if $f_1, \ldots, f_n$ is a generating set, then $\pi_i(f_j)$ for $i \in \mathbb{Z}$, $j = 1, \ldots, n$ is a finite generating set also.
I don't think it especially matters what the grading monoid is.
Note that you can have submodules of a graded submodule which are not graded modules,for example the ideal $(x + y^2)$ in $k[x,y]$. This ideal cannot be generated by homogeneous forms (much less finitely many), because that would make it a graded submodule (and it would have to contain $x$, but it clearly doesn't).