# Definition:Tensor Product of Modules as Abelian Group

## Definition

Let $R$ be a ring with unity.

Let $M$ be a unitary right module and $N$ a unitary left module over $R$.

### Definition 1: by universal property

Their tensor product is a pair $(M \otimes_R N, \theta)$ where:

satisfying the following universal property:

For every pair $(P, \omega)$ of an abelian group and an $R$-balanced mapping $\omega : M \times N \to P$, there exists a unique group homomorphism $f : M \otimes_R N \to P$ with $\omega = f \circ \theta$.

### Definition 2: direct construction

Their tensor product is the pair $(M \otimes_R N, \theta)$, where:

### Definition 3: construction via abelian groups

Let $(M \otimes_\Z N, u)$ be the tensor product of their underlying abelian groups.

The tensor product of $M$ and $N$ is the the pair $(M \otimes_R N, \theta)$, where: