Homogeneous ideal

This is an analogue in graded rings of the ring subset property: 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 $d^{t h}$ 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
$I = ⨁_{d = 0}^{\infty} I \cap A_{d}$ . The intersection $I \cap A_{d}$ is denoted $I_{d}$ , and is the $d^{t h}$ graded component of $I$ , viewed as a graded $A$ -module.