Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module. Let $I$ be a proper ideal of $R$ and $b\notin I$ be such that the maps $(I+Rb) \otimes_R M \to R\otimes_R M$ and $(I:b) \otimes_R M \to R \otimes_R M$ are injective. Then is the map $I \otimes_R M \to R \otimes_R M$ injective ?
If this is not true in general, then what if we assume $R$ is a domain or assume $M$ is finitely generated ?