Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
If $I$ and $J$ are gradedmonomial ideals, then the question is true.
If $I$ and $J$ are graded ideals, then the question is true.
If $I$ and $J$ are monomial ideals, then the question is true.