Relative tensor product

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Let A be a right R-module and B be a left R-module (see the definition of left and right module in module (mathematics)). The relative tensor product A\otimes_R B is defined to be the quotient

A\otimes B/I

where I is the ideal generated by all elements

x\cdot r\otimes y-x\otimes r\cdot y\quad (x\in A, y\in B, r\in R)


If A is a left S-module and the left action of S commutes with the right action of R on A, then A\otimes_R B is naturally a left S-module.

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