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  "body": "\n## Summary\n\nA bimodule over two rings $R$ and $S$ is a left [[module|0NZ8JA25WK]] over $R$ that is also a right module over $S$.\nThe two actions must commute.\nBimodules are the natural objects between two rings: tensoring with an $(R, S)$-bimodule turns right $R$-modules into right $S$-modules.\nThe two-sided ideals of a ring are exactly its sub-bimodules.\nBimodules underlie Morita theory and Hochschild cohomology.\n\n## Explanations\n\n### L1 human\n\nA bimodule is acted on from both sides, by one ring on the left and by another ring on the right.\nThe only condition linking the two actions is that it does not matter which side acts first.\nRectangular $m \\times n$ matrices are the model: square $m \\times m$ matrices multiply them on the left, square $n \\times n$ matrices multiply them on the right, and $(AX)B = A(XB)$.\nEvery ring is a bimodule over itself, with multiplication on both sides.\n",
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