tensor products of modules over an algebra

DEFINITION 9.1   Let $A$ be a (not necessarily commutative) ring. Let $M$ be a right $A$-module. Let $N$ be a left $A$-module. Then we define the tensor product of $M$ and $N$ over $A$, denoted by

$$M\otimes_A N$$

as a module generated by symbols

$$\{ m \otimes n ; m\in M, n\in N\}$$

with the following relations.

1. $$(m_1 + m_2)\otimes n =m_1 \otimes n + m_2 \otimes n \quad(m_1,m_2 \in M,\ n\in N)$$

2. $$m\otimes (n_1 + n_2) =m \otimes n_1 + m \otimes n_2 \quad(m\in M,\ n_1,n_2 \in N)$$

3. $$m a \otimes n = m \otimes a n \qquad (m\in M,\ n\in N,\ a \in A)$$