Definition quadratic forms over a module

I have a question about the following definition from my book:

Quote:

Originally Posted by **definition**

Let V be a module over a commutative ring A. A function Q: V -> A is called a quadratic form on V if the following properties are satisfied:

(1)

for all

and

(2)

is a bilinear form.

My questions are the following:

- When I looked up the definition of a quadratic form over a field property I saw that (1) is sufficient. Why should property (2) in this case also be included in the definition? Can anyone give an example of a function that is not a quadratic form, but for which (1) holds?