If $S$ is a graded ring, for which $k \in \mathbb{Z}$ do we have $0 \in S_k$?
I think there shouldn't exist such a k. So as 0 is the empty sum we don't need this?
2026-03-26 18:30:02.1774549802
$0 \in S_k$ for which k?
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A $\mathbb{Z}$-graded ring $S$ is, as abelian group with respect to addition, $$ S=\bigoplus_{k\in\mathbb{Z}}S_k $$ where the $S_k$ are additive subgroups with the further property that $$ S_hS_k\subseteq S_{h+k} $$ as regards to multiplication.
Thus $0\in S_k$ for every $k$.