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In the Wikipedia page it says if $I$ is a homogeneous ideal in graded ring $A=\oplus_{n\ge0}A_n$, then $\frac{A}{I}$ is a graded ring decompose as:

$\frac{A}{I}=\oplus_{n\ge0}\frac{A_i+I}{I}$

I wonder where we use the condition homogeneous? I think as long as $I$ is an ideal then we have $(A_i+I)(A_j+I)\subset A_{i+j}+I$ so $A/I$ is a graded ring.

But both Wikipedia page and my book highlight the condition homogeneous so I wonder if I misunderstand some concept.

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1 Answer 1

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An ideal $\mathfrak{a}$ of $S$ is called homogeneous if it satisfies the condition:

If $a\in \mathfrak{a}$ and $a=\sum_da_d$ with $a_d\in S_d$, then each $a_d\in \mathfrak{a}$.

Suppose $I$ is a homogeneous ideal, $a_i\in A_i,a_j\in A_j$ with $i\neq j$ such that $a_i+I=a_j+I$ , then

$a_i-a_j\in I$, hence $a_i,a_j\in I$ by the above condition, therefore $a_i+I=a_j+I=0$, so we have $\frac{A_i+I}{I}\cap \frac{A_j+I}{I}=0$ with $i\neq j$.

If $I$ is not homogeneous, for $i\neq j$, $\frac{A_i+I}{I}\cap \frac{A_j+I}{I}=0$ may not be always true.

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