Definition 10.11.1. Let $R$ be a ring, $M, N, P$ be three $R$-modules. A mapping $f : M \times N \to P$ (where $M \times N$ is viewed only as Cartesian product of two $R$-modules) is said to be $R$-bilinear if for each $x \in M$ the mapping $y\mapsto f(x, y)$ of $N$ into $P$ is $R$-linear, and for each $y\in N$ the mapping $x\mapsto f(x, y)$ is also $R$-linear.
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