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Let $R$ and $S$ be commutative rings over a field $k$. Let $I$ be an ideal of the tensor ring $R\otimes_{k} S$. It is true that there exist ideals $I_{1}$ and $I_{2}$ of $R$ and $S$ respectively such that $$ I=I_{1}\otimes_{k} I_{2}? $$ If this is not true, are there any description of $I$? What if we don't assume commutativity of one of rings?

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