Let $R$ be a commutative ring and $I,J$ ideals of $R$. Is true that $I \otimes_R J \simeq IJ $ ?
If not, have conditions on $I,J$ such that it is true?
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Yes, but I want to know if have conditions over $I,J$ that makes this result true – pmtm Sep 22 '21 at 00:42
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For example, if $J=M$ is a flat module, the result is true – pmtm Sep 22 '21 at 00:42
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In general, exists a condition over $I,J$? – pmtm Sep 22 '21 at 00:43
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4The given answer proves that the map $I \otimes_R J \to IJ, , a \otimes b \mapsto ab$ is bijective if and only if $\operatorname{Tor}_1^R(R/I,J)=0$. – azif00 Sep 22 '21 at 00:45