## Lecture Notes 6: Addition on Lines

**Def**: A *division ring (skewfield)* is a set (D, +, ×) with two binary operations (addition and multiplication) such that (D,+) is an abelian group with identity 0, and (D-{0}, ×) is a group with identity 1, and multiplication distributes over addition.
A division ring in which (D-{0}, ×) is an **abelian** group is a field. We shall see examples of division rings which are not fields later. There are no finite examples of this type since it can be shown that:

**Wedderburn's Theorem**: Every finite division ring is a field.

We will introduce the concepts of addition and multiplication of points on a line in an affine plane (with respect to two fixed points O and I) and show that if the affine plane satisfies an additional property (the Desargues theorem), then the set of points on the line with these binary operations forms a division ring.

### Addition on Lines

*Definition of Addition*:
Let *l* be a fixed line in an affine plane and fix a point O on *l*. To add the points A and C on *l* (with respect to O):

- Select a point B not on
*l*.
- Let
*m* be the line through B parallel to *l*.
- Let
*k* be the line through A parallel (or equal) to the line through O and B.
- Let D =
*m**k*.
- Let
*n* be the line through D parallel (or equal) to the line through B and C.
- Define A + C =
*n**l*.

**Theorem**: O + C = C for any point C on *l*.
**Theorem**: A + O = A for any point A on *l*.

**Corollary**: O + O = O.

Thus, the point O acts as an additive identity (zero) for what will be a division ring. To prove that this addition satisfies the other needed properties, we will have to introduce another property for our affine plane.