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For $R$-modules $M,N$, the tensor product of modules is an $R$-module $M{\otimes }_{R}N$ together with the balanced mapsuch that every balanced map $b:M\times N\to P$ factors through one unique $R$-linear map $\tilde{b}$:Equivalently,naturally in $P$. This is the universal property of the tensor product of modules.

The Tensor-hom adjunction is the natural isomorphismFor an R-module homomorphism $f$, it is given explicitly byConversely, an $R$-linear map $g:M\to {\mathrm{Hom}}_R(N,P)$ determines the balanced map $(m,n)\mapsto g(m)(n)$, so the universal property of the tensor product of modules givesThese formulas are inverse to each other because pure tensors generate the tensor product of modules.

The tensor product of an $R$-module $M$ and an $R$-module $N$ is an $R$-module $M{\otimes }_{R}N$ generated by pure tensors $m\otimes n$, subject to additivity in both variables and the balancing relation

Every balanced map $b:M\times N\to P$ factors uniquely through an $R$-module homomorphism $\tilde{b}:M{\otimes }_{R}N\to P$ satisfying $\tilde{b}(m\otimes n)=b(m,n)$.