Tensor Products of Modules

Balanced Map

Tensor Products of Modules

Suppose $M$ is a left $R$-module and $N$ is a right $R$-module. The tensor product of $M$ and $N$ over $R$, denoted by $M{\otimes }_{R}N$, is an abelian group satisfying the following universal property: for any abelian group $A$ and any $R$-balanced map $f:M\times N\to A$, there exists a unique group homomorphism $\tilde{f}:M{\otimes }_{R}N\to A$ such that the following diagram commutes:

In other words, it is the free module generated by $M\times N$ modulo the relations

• $(m_1+m_2,n)\sim (m_1,n)+(m_2,n)$,
• $(m,n_1+n_2)\sim (m,n_1)+(m,n_2)$,
• $(m\cdot r,n)\sim (m,r\cdot n)$,

that make the map $M\times N\to M{\otimes }_{R}N$ bilinear.