homogeneous ideal

Let $R = \oplus_{g\in G} R_g$ be a graded ring. Then an element $r$ of $R$ is said to be homogeneous if it is an element of some $R_g$ . An ideal $I$ of $R$ is said to be homogeneous if it can be generated by a set of homogeneous elements, or equivalently if it is the ideal generated by the set of elements $\bigcup_{g\in G} I\cap R_g$ .

One observes that if $I$ is a homogeneous ideal and $r=\sum_i r_{g_i}$ is the sum of homogeneous elements $r_{g_i}$ for distinct $g_i$ , then each $r_{g_i}$ must be in $I$ .

To see some examples, let $k$ be a field, and take $R=k[X_1,X_2,X_3]$ with the usual grading by total degree. Then the ideal generated by $X_1^n+X_2^n-X_3^n$ is a homogeneous ideal. It is also a radical ideal. One reason homogeneous ideals in $k[X_1,\ldots,X_n]$ are of interest is because (if they are radical) they define projective varieties; in this case the projective variety is the Fermat curve. For contrast, the ideal generated by $X_1+X_2^2$ is not homogeneous.