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:


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

is a left module.

Axiom 1
Let $x, y \in T$ with $x = \family {s_i}_{i \mathop \in I}$ and $y = (t_i)_{i \mathop \in I}$.

Let $\lambda\in R$.

Then:

Axiom 2
Let $x \in T$ with $x = \family {s_i}_{i \mathop \in I}$

Let $\lambda, \mu \in R$.

Then:

Axiom 3
Let $x\in T$ with $x = \family {s_i}_{i \mathop \in I}$.

Let $\lambda, \mu \in R$.

Then: