I know that in a graded domain, if a homogeneous element is written as a product of two elements, then each of two elements is also homogeneous. That is, the set of all the homogeneous elements of domain is saturated.
But I don't know if this argument also holds for ideal cases. For example, let $D$=$\bigoplus$$_{\alpha}$$D_{\alpha}$ be a graded domain. If a homogeneous ideal $I$ can be written as a product of two ideals, say $I=AB$, then is each of them also homogeneous?
Thank you for answering, in advanced.