# Bilinear Forms over Rings

Quadratic forms were probably first studied by Babylonian mathematicians and from its birth until the beginning of the twentieth century, quadratic forms were only studied over the field of real numbers, the complex field or the ring of integers. It was only in 1937 that Ernst Witt published a groundbreaking paper that would become the founding stone to incorporate the study of quadratic forms into the theory of pure algebra.

We want to embark on a journey to study quadratic (and hermitian) forms on modules over general rings, but first, we will give an introduction into the theory of bilinear forms over rings (our rings will always be associative with identity element 1).

## Bilinear Forms

### Definition I.1.1

Let *bilinear form* on *symmetric*, if *screw-symmetric*, if *symplectic* or *alternating*, if

An object *symmetric bilinear module* over

### Remark

A symplectic form is always screw-symmetric, for if

### Remark

If

### Definition and Remark I.1.2

Now let *isometry* and we say that the two bilinear modules are *isometric* if there exists a bijective homomorphism between them. The relation of being isometric is an equivalence relation, and thus it makes sense to say that a bijective isometry is an isomorphism in the category of bilinear modules. (We will talk more about that category in a later entry.)

### Definition I.1.3

Two elements

### Remark

It is easy to see that

### Definition I.1.4

A sum of submodules *internal orthogonal sum*, or orthogonal sum, if it is a direct sum, that is, if

The *external orthogonal sum* of two submodules

### Remark

Those two types of orthogonal sums are obviously canonically isomorphic.

### Definition and Remark I.1.5

For a given bilinear module

### Remark

We thus have a module isomorphism between the set of all bilinear forms on

The next definition is essential to further specify the bilinear forms we are actually interested in:

### Definition I.1.6

A bilinear module *non-degenerate*, if *regular*, if

### Remark

For finite vector spaces over fields those two definitions are the same, as then

### Remark

It is easy to see that

### Remark

In other words: Regularity means that for a given

We will now specify the modules we are truly interested in:

### Definition I.1.7

A

### Definition I.1.8

A bilinear module *regular bilinear module* if *bilinear space* if additionally

### Remark

If

### Theorem I.1.9

An orthogonal sum of bilinear modules is non-degenerate / regular if and only if each summand is a non-degenerate / regular bilinear module.

As a last result for this introductory chapter, we want to state an easy to prove theorem over fields:

### Theorem I.1.10

Let

, , and is non-degenerate.

## Literature

- Symmetric Bilinear Forms by J. Milnor and D. Husemoller
- Arithmetic Theory of Quadratic Forms by R. Scharlau