= Homogeneous element of a graded ring
{title2=$a\in R_d$}
= Homogeneous elements of a graded ring
{synonym}
In a <graded ring> $R=\bigoplus_{d\geq0}R_d$, a homogeneous element of degree $d$ is an element of the component $R_d$. Every <ring> element is a finite sum of homogeneous components. A nonzero homogeneous element has a unique degree because the sum is direct; zero belongs to every component and usually has no unique assigned degree.
In $k[x,y]$ with total-degree grading, $x^2+3xy-y^2$ is homogeneous of degree two, although it has three terms. The element $x^2+y$ is not homogeneous: its nonzero components lie in degrees two and one. The grading rule gives $R_dR_e\subseteq R_{d+e}$, so a nonzero product of homogeneous elements has degree $d+e$.
Grading depends on the declared convention. Assigning weight two to $x$ and weight one to $y$ makes $x+y^2$ homogeneous of weighted degree two. Homogeneity organizes computations in <graded modules> and the equations used in <projective geometry>.
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