 Homogeneous element of a graded ring

In a graded ring $R={⨁}_{d\ge 0}{R}_d$, 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 $R_d{R}_e\subseteq R_{d+e}$, 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.