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universality of tensor products

DEFINITION 3.2   Let $ A$ be a (not necessarily commutative) ring. Let $ M$ be a right $ A$ -module. Let $ N$ be a left $ A$ -module. Then for any module $ X$ , a map $ f:M\times N \to X$ is said to be an $ A$ -balanced biadditive map if it satisfies the following conditions.
  1. $ f(m_1+m_2,n)=f(m_1,n)+f(m_2,n)
\quad (\forall m_1,m_2 \in M, \forall n \in N)
$
  2. $ f(m,n_1+n_2)=f(m,n_1)+f(m,n_2)
\quad (\forall m \in M, \forall n_1,n_2 \in N)
$
  3. $ f(m a ,n)=f(m, a n)
\quad (\forall m \in M, \forall n \in N, \forall a \in A)
$

LEMMA 3.3   Let $ A$ be a (not necessarily commutative) ring. Let $ M$ be a right $ A$ -module. Let $ N$ be a left $ A$ -module. Then for any module $ X$ , there is a bijective additive correspondence between the following two objects.
  1. An $ A$ -balanced bilinear map $ M\times N \to X$
  2. An additive map $ M\otimes_A N \to X$



2007-12-11