Can we always choose the generators of an ideal of a Noetherian ring to be homogeneous? 
Let $R$ be a $k$-subalgebra of $S=k[x_1,x_2,\dots,x_n]$. Let $m\subseteq R$ be the ideal generated by the homogeneous elements of $R$ of positive degree. As $S$ is Noetherian, the ideal $mS$ has a finite set of generators, which may be chosen to be homogeneous elements of $m$. 

Why can we choose homogeneous elements of $m$ to be the generators? 

Elements in $mS$ are $S$-linear combinations of elements in $m$. Take generators of $mS$ and split up each linear combination to obtain generators $m_i \in m$ of $mS$. By definition of $m$, each $m_i$ is an $R$-linear combination of homogeneous elements of $R$. Splitting these up again, gives the desired generators.
