Let $M$ be a right $R$-module and $N$ a left $R$-module. In defining trnsor product $M\otimes_R N$, Curtis-Reiner in the book "Repre. of Finite Groups Assoc. Algebra" mentions following:
We shall have to consider mappings that are somewhat weaker than the bilinear mappings, and following Chevalley, we call them the balanced mappings in the following sense:
If $P$ is any additive abelian group, a mapping $f:M\times N\rightarrow P$ is said to be balanced if
1) $f(m,n_1+n_2)=f(m,n_1)+f(m,n_2)$
2) $f(m_1+m_2,n)=f(m_1,n) + f(m_2,n)$.
3) $f(m,rn)=f(mr,n) \,\,\,\,\,\,\,\,\,\,\,\,(m's \in M, n's\in N, r\in R)$
Q. In what sense, the balanced maps are weaker than the bilinear mappings?
Best Answer
Usually, bilinearity involves an $R$-module structure on $P$ and a condition such as $f(mr, n) = f(m,rn) = rf(m,n)$.
For instance this is what one might require when defining the tensor product over a commutative ring.