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? 

This is true because $m$ is generated by homogeneous elements, and $mS$ is generated by $m$. Any finite generating set of $mS$ will consist of elements that are linear combinations of elements of $m$ with coefficients in $S$. If the generating set has elements not in $m$, then they'll be of the form
$$\sum{s_im_i}$$
Instead of using this element we can replace it with the finitely many elements $m_1,m_2,\ldots$ and still have a generating set. Similarly, since $m$ is generated by homogeneous elements, if these elements of $m$ are not homogenous we can replace them with homogeneous elements of which they are a linear combination.
