Let $R$ be a ring, not necessarily unital or commutative. Let $I$ and $J$ be nontrivial two-sided ideals of $R$. Their product is defined as $IJ:=\{ i_1j_1+\dots+i_lj_l:l\ge 1, i_n\in I, j_n \in J, n\in\mathbb{N} \}$, which can be proved to be a two-sided ideal of $R$ as well.
Question. Is it correct to claim $IJ=I\otimes_R J$? Here we are treating $I$ and $J$ as two-sided $R$-modules, and using the definition of tensor products of $R$-modules.
Perhaps they are isomorphic only as $R$-modules, but the tensor product loses the ring structure (multiplication)? Thanks in advance.
