Suppose I have a ring $R$, and I want to find a subset $J \subset R$ such that $$xy = 0 : x,y \in J$$ (and hopefully, find the 'maximal' such subset $J$). Is there a name for such subsets $J$, or are they all just trivial?
2026-04-08 05:22:47.1775625767
Subring such that all products are zero
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Take $R=\mathbb{Z}/4$, represented by $0,1,2,3$. Then $J=\{0,2\}$ is such a set. The sets $$ \mathrm {Ann} _{R}(S)=\{r\in R\mid \forall s\in S,rs=0\} $$ are called (left) annihilator sets.