Finding some homogeneous generators of an ideal.

Question

Suppose that $\mathfrak a$ is an homogeneous ideal of $K[T_1,\ldots, T_n]$ where $K$ is a field of characteristic $0$ and $T_1,\ldots,T_n$ are indeterminates. Moreover suppose that $\mathfrak a$ has a system of generators in $L[T_1,\ldots, T_n]$ where $L\subseteq K$.

Can I find a system of homogeneous generators in $L[T_1,\ldots, T_n]$?

thanks in advance

Answer 1

Yes, an ideal $I$ in a polynomial ring is homogeneous iff for every element $f$ in the ideal, each of the homogeneous parts of $f$ also lie in $I$. The homogeneous parts of the given generators then perform as required.