Assume $R$ is a ring with an element $r$. Assume $R/rR$ is flat as $R$-module. I would like to show that there is an exact sequence $r^2R \rightarrow rR \rightarrow rR \otimes_R (R/rR)\rightarrow 0$. There is obviously an exact sequence $0 \rightarrow rR \rightarrow R \rightarrow R/rR \rightarrow 0$ considering which I showed that $rR \otimes_R (R/rR) = 0$ but I am stuck on this following step. Can anyone give directions? Thanks in advance
2026-04-04 09:44:52.1775295892
Sequence of modules
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Hint:
Tensoring (on the left) with $rR$ yields a sequence $$0\longrightarrow rR\otimes_R rR\longrightarrow rR\otimes_R R\longrightarrow rR\otimes_R R/rR\longrightarrow 0,$$which is exact because $R/rR$ is a flat $R$-module. Remember that $\; rR\otimes_R R\simeq rR$. With this identification, what is the image of $ rR\otimes_R rR$ in $rR\mkern1mu$?