Why is it that an ideal is homogeneous if and only if it is generated by homogeneous elements? Hartshorne says the following on pg. 9 

An ideal is homogeneous if and only if it is generated by homogeneous elements.

Take $\langle x+y,x^3+y^3\rangle$. It is generated by homogeneous elements. But how is it homogeneous? Clearly $x+y+x^3+y^3$ is not homogeneous.

If an ideal I is homogeneous, and if a possibly non-homogeneous P(x,y) element belongs to I, each homogeneous component of P(x,y) belongs to I. Therefore, given any system of generators of $I$, each of their homogeneous components belo,gs to $I$ – in other words, $I$ is generated by the homogeneous components of a system of not necessarily homogeneous set of generators.
