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Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, let $tr_R(M):=Im(\tau_M)$, where $\tau_M:M\otimes Hom(M,R)\to R$ is the map defined as $\tau_M(m\otimes f)=f(m)$.

Let $I$ be a proper ideal of $R$ and $b\in R$ be such that $I+Rb$ and $(I:b)$ are isomorphic (as an $R$-module) to the trace of some finitely generated $R$-modules. Then is $I$ also isomorphic to the trace of some finitely generated $R$-module ?

If this is not true in general, then what if we also assume $R$ is local ? An integral domain ?

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