Homogeneous ideal

Definition

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 $I\leq A$ 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.