Graded ring, and its homogeneous ideals : $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $

Question

Let $ B = \displaystyle \bigoplus_{n \in \mathbb {Z}} B_n $ be a graded ring. Let $ I $ be an ideal of $ B $. Why is $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $ equivalent to $ I $ has a generating family consisting of homogeneous elements ? What does mean that : $ I $ has a generating family consisting of homogeneous elements?

Thank you in advance.

Answer 1

To spell out explicitly: "$I$ has a generating family consisting of homogeneous elements" means

There exists a set $\{b_\lambda\} \subseteq B$, with each $b_\lambda$ homogeneous (i.e. $b_\lambda \in B_n$ for some $n$), such that $I = (b_\lambda)$ (i.e. every $x \in I$ can be written as $x = \sum_\lambda r_\lambda b_\lambda$ with $r_\lambda \in B$, and only finitely many $r_\lambda \ne 0$).

That this is equivalent to saying $I = \bigoplus_n (I \cap B_n)$ follows directly from the definition, by grouping $\displaystyle x = \sum_\lambda r_\lambda b_\lambda = \sum_n \left(\sum_{\deg b_\lambda = n} r_\lambda b_\lambda \right)$