Let $R$ be a ring with identity with a 2 sided ideal $I$ and a left ideal $L$ such that $I\subseteq L$. Must there exist a left $R$-module $M$ and an element $m_0$ of $M $ such that $\mathrm{Ann}(m_0)=L,\mathrm{Ann}(M)=I$ ?
Thank you.
2026-03-29 23:29:14.1774826954
Is there any relationship between the $\mathrm{Ann} (M) $ and $\mathrm{Ann} (m_0) $ (other than that $\mathrm{Ann}(M)\subseteq \mathrm{Ann}(m_0) $)?
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Sure, take $M=R/I\oplus R/L$ and $m_0=(0,1)$.