LEC ALG4 14

[!Proposition]
Let $A$ be a graded ring $A=A_0\oplus A_1\oplus \dots$. Let $I$ be an $A$ ideal. TFAE:

1. $I$ is generated by homogeneous elements.
2. $I={⨁}_{n⩾0}I\cap A_n$ ¹

Such an ideal is called a homogeneous ideal.

[!Proof]-

$(1⟹2)$ by definition.

$(2⟹1)$ Let $Z=\{{\zeta }_{\lambda }:\lambda \in \Lambda \}$ be homogeneous such that $I$ is the ideal generated by $Z$. Let $\eta \in I$. Write $\eta ={\sum }_{i=1}^r{a}_i{\zeta }_i$ with $a_i\in A$. For any $b\in A$, we can write $b={\sum }_{i⩾0}{b}_i$ with $b_i\in A_i$ and $b_i=0$ for all but finitely many $i$.

Footnotes

1. Elements of $A_n$ are called homogeneous of degree $n$. $(2)$ says that $I={\sum }_{n⩾1}I\cap A_n$. ↩