A basic property of graded rings

Question

I've read that if $I$ is a homogeneous ideal of a graded ring $R$, then $R/I$ is also a graded ring. I'm unclear why we need the homogeneous condition on $I$. Is it to avoid confusion on the degree of homogeneous elements in the quotient?

Answer 1

Yes, that's exactly the reason.

Any sensible grading on $R/I$ should make the quotient homomorphism $\pi: R \to R/I$ a graded homomorphism, i.e. degree-preserving. Such a grading, if it exists, is therefore uniquely determined: The degree $d$ elements of $R/I$ are the images of degree $d$ elements of $R$ under $\pi$. This does give a well-defined grading iff the sum $R/I = \sum_d \pi(R_d)$ is direct, where $R_d$ denotes the degree $d$ elements of $R$. And this amounts precisely to the condition that $I$ be a homogeneous ideal: If $f_d \in R_d$ such that $\sum f_d \in I$, then $f_d \in I$ for all $d$.