Definition of homogeneous ideal I'm a little confused about the definition of a homogeneous ideal. I have the following two definitions:


*

*An ideal $I\subset k[X_{0}, \dots, X_{n}]$ is homogeneous if $I$ is
generated by (finitely many) homogeneous polynomials.

*An ideal $I\subset k[X_{0}, \dots, X_{n}]$ is homogeneous if $I$ can 
be generated by homogeneous polynomials.
So does a homogeneous ideal have to be an ideal generated only by finitely many homogeneous polynomials or do we allow the infinite case?
I know that every ideal can be generated by finitely many when $k$ is a field, but an ideal, which we have generated by an infinite number of homogeneous polynomial, is not necessarily generated by a finite number of homogeneous polynomials.. right?

To clarify my last sentence: I had an exercise where $I\subset k[x_{1}, \dots, x_{n} ]$ is an ideal and $I^{h}$ was the ideal generated by $\lbrace f^{h};f\in I \rbrace$ where $f^{h}$ is the homogenization of $f$ i.e. it is a homogeneous polynomial. 
The exercise was to show that $I^{h}$ is a homogeneous ideal. From the second definition above, this is "obvious" a homogeneous ideal (since it is generated by homogeneous polynomials). But from the first definition we actually have to show that it can be generated by a finite number of homogeneous polynomials. The solution I was given, was not a trivial solution.

In a notherian ring, an ideal with an infinite set of generators is generated by some finite subset of those generators (just keep taking more non-redundant ones until Emmy tells you to stop).  So this distinction is not important.  If we have generators that are homogeneous, then we have a finite set of generators that are homogeneous.
Maybe I'm missing something, but if somebody solved the problem you mention in a non-trivial way, it's possible that they were inadvertently re-proving something like the Hilbert basis theorem.
