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. \( B(x, a \bullet y) = a * B(x, y)\)

for all \(a \in R, x, y, z \in M\).

Let \(R\) be a commutative ring, \(M\) a \(R\)-module. An quadratic form \(Q\) over \(M\) is a map \(Q : M \to R\), satisfying:

1. \( Q(a \bullet x) = a * a * Q(x)\) for all \(a \in R, x \in M\).

2. there exists a companion bilinear form \(B : M \to M \to R\), such that \(Q(x + y) = Q(x) + Q(y) + B(x, y)\)

In some literatures, the bilinear form is denoted \(\Phi \), and called the polar form associated with the quadratic form \(Q\), or simply the polar form of \(Q\).