## 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.

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):

1. Select a point B not on l.
2. Let m be the line through B parallel to l.
3. Let k be the line through A parallel (or equal) to the line through O and B.
4. Let D = mk.
5. Let n be the line through D parallel (or equal) to the line through B and C.
6. Define A + C = nl.
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.