definition. Bilinear form [ca-0003]

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BilinForm

L∃∀N

Let \(R\) be a ring, \(M\) an \(R\)-module.

An bilinear form \(B\) over \(M\) is a map \(B : M \to M \to R\), satisfying:

1. \( B(x + y, z) = B(x, z) +B(y, z) \)
2. \( B(x, y + z) = B(x, y) +B(x, z) \)
3. \( B(a \bullet x, y) = a * B(x, y)\)
4. \( B(x, a \bullet y) = a * B(x, y)\)