# Definition:Tensor Product of Modules as Abelian Group

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## 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:

- $M \otimes_R N$ is an abelian group
- $\theta : M \times N \to M \otimes_R N$ is an $R$-balanced mapping

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:

- $M \otimes_R N$ is the quotient group of the free $R$-module $R^{(M\times N)}$ on the direct product $M \times N$, by the subgroup generated by the set of elements of the form:
- $(m_1 + m_2, n) - \lambda (m_1, n) - (m_2, n)$
- $(m, n_1 + n_2) - \lambda (m, n_1) - (m, n_2)$
- $(m \cdot \lambda, n) - (m, \lambda \cdot n)$

- for $m, m_1, m_2 \in M$, $n, n_1, n_2 \in N$ and $\lambda \in R$, where we denote $(m, n)$ for its image under the canonical mapping $M \times N \to R^{(M\times N)}$.

- $\theta : M \times N \to M \otimes_R N$ is the composition of the canonical mapping $M \times N \to R^{(M\times N)}$ with the quotient module epimorphism $R^{(M\times N)} \to M \otimes_R N$.

### 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:

- $M \otimes_R N$ is the quotient of $M \otimes_\Z N$ by the subgroup generated by the elements of the form:
- $u(m \cdot \lambda, n) - u(m, \lambda \cdot n)$

- $\theta : M \times N \to M \otimes_R N$ is the composition of $u$ with the quotient group epimorphism $M \otimes_\Z N \to M \otimes_R N$.