homogeneous ideal HomogeneousIdeal 2001-10-15 18:35:49 2004-02-16 00:30:32 Definition graded ring /hoh-moh-gee''-nee-uhs/ homogeneous homogeneous element commutative algebra algebraic geometry \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $R = \oplus_{g\in G} R_g$ be a graded ring. Then an element $r$ of $R$ is said to be \emph{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 \PMlinkname{Fermat}{FermatsLastTheorem} curve. For contrast, the ideal generated by $X_1+X_2^2$ is not homogeneous.