Homogeneous ideal

Symbol-free definition
An ideal in a graded ring is termed a homogeneous ideal or a graded ideal if it satisfies the following equivalent conditions:


 * It is generated by homogeneous elements
 * It equals the sum of its intersections with all the homogeneous components (or graded components)
 * It is a graded submodule of the graded ring, viewed as a graded module over itself

Definition with symbols
Let $$A$$ be a graded ring where the $$d^{th}$$ graded component is denoted $$A_d$$. Then, an ideal $$I \le A$$ is termed a homogeneous ideal or graded ideal if it satisfies the following conditions:


 * $$I$$ is generated (as an $$A$$-module) by homogeneous elements
 * $$I = \bigoplus_{d=0}^\infty I \cap A_d$$. The intersection $$I \cap A_d$$ is denoted $$I_d$$, and is the $$d^{th}$$ graded component of $$I$$, viewed as a graded $$A$$-module.