What is considered to be the natural (injective) homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$? Let $R$ be a ring and $I,J,L \unlhd R$ such that $J \subseteq I$. > What is considered to be the natural homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$ ? > > Remark: It must be injective! Will $x + IL \mapsto x \otimes L$ or $x+IL \mapsto 0 \otimes (x+L)$ work? We also have $x+IL \mapsto x \otimes (x+L)$ but this isn't a homomorphism. What can I take here? Thanks.

Recall for an $R$-module $M$ and $\mathfrak{a}$ ideal of $R$ the canonical isomorphism $$M \otimes_R R/\mathfrak{a} \cong M/\mathfrak{a}M$$ In particular $$I \otimes R/L \cong I/IL$$ since $IL+J \subset I$, you have the inclusion $$\frac{IL+J}{IL} \to \frac{I}{IL} \cong I \otimes \frac{R}{L}$$