Tensor Product is Module

Theorem
Let $R$ be a ring.

Let $M$ be a $R$-right module.

Let $N$ be a $R$-left module.

Then


 * $T = \displaystyle \bigoplus_{s \mathop \in M \mathop \times N} R s$

is a module

Axiom 1
Let $x,y\in T$ with $x=\{s_i\}$ and $y=\{t_i\}$

Let $\lambda\in R$

Then

$\lambda\circ(x+y)=\lambda\circ\{s_i+t_i\}=\{\lambda\circ s_i+\lambda\circ t_i\} = \lambda\circ \{s_i\}+\lambda\circ \{t_i\}=\lambda\circ x + \lambda\circ y$

Axiom 2
Let $x\in T$ with $x=\{s_i\}$

Let $\lambda,\mu\in R$

Then

$(\lambda+\mu)\circ x=(\lambda+\mu)\circ\{s_i\}=\{\mu\circ s_i+\lambda\circ s_i\} = \lambda\circ x + \lambda \circ \mu$

Axiom 3
Let $x\in T$ with $x=\{s_i\}$

Let $\lambda,\mu\in R$

Then

$(\lambda\times\mu)\circ x=(\lambda\times\mu)\circ\{s_i\}=\{(\lambda\times\mu)\circ s_i\} = \{\lambda\circ(\mu\circ s_i)\}=\lambda\circ\{\mu\circ s_i\}=\lambda\circ (\mu\circ x)$